As far as their role in politics and economics, the one who really started to turn the tide against Gilded Age laissez-faire chaos was actually McKinley, under whom Roosevelt had served as VP before the President's assassination by a bitter son-of-immigrants.  And like Trump, McKinley was not as enthusiastically Progressive as Roosevelt. At the very beginning of the reversal, populism is going to be moderate and tempered.
Since Trump hasn't been elected yet, it's too early to make detailed comparisons, so I'll just run through a quick list of similarities. (Quotes are from Wikipedia.)
First, both ran for President during a historical realignment of the party system. For McKinley, it was the shift into the Fourth Party System, or the Progressive Era, after the bitterly divisive Civil War and Reconstruction era. In Trump's case, it is out of the neoliberal / neoconservative Sixth Party System that has seen growing polarization, class war, and intra-elite competition since the mid-1970s.
Second, both campaigned on economic nationalism and making tariffs a respectable topic and tactic.  McKinley rode into the House of Representatives in the late 1870s by calling for protective tariffs, and by the end of his Congressional career, had achieved the McKinley Tariff of 1890. After assuming the presidency in 1897, he followed up by championing the Dingley Act, which restored tariffs after their moderate reduction in 1894 (under the Wilson-Gorman Tariff). By the 1909 Payne-Aldrich Tariff Act, the trend he began would culminate in a deep split within the Republican Party, peeling off the laissez-faire Old Guard from the insurgent nationalist Progressives.
Trump will not cause such a fundamental rift himself, since that takes some time to grow. But he will certainly be the first to inject economic nationalism into a Republican Party deeply steeped in laissez-faire ideology. And like McKinley, he will be advocating tariffs on their ability to strengthen domestic industries, rather than as a way for government to raise more money for itself. They see them as deterrents against economic imperialism from foreign industries -- they hope to never actually collect the tariffs because, in our case, the Chinese will think twice about dumping all their cheap crud in our country to begin with. The goal is to start making quality stuff here in America, thereby providing good jobs to those who produce the stuff, and using those higher incomes to afford quality stuff.
Third, both aimed to reconcile workers and business owners. As a lawyer before taking office, McKinley successfully defended (pro bono) a group of miners who had gotten into a fight with strikebreakers. That was in 1876, long before labor rights were popular, let alone among Republicans. This continued when he became the Governor of Ohio:
Although McKinley believed that the health of the nation depended on that of business, he was evenhanded in dealing with labor. He procured legislation that set up an arbitration board to settle work disputes and obtained passage of a law that fined employers who fired workers for belonging to a union.
This was during the 1890s, when politicians were beginning to tolerate labor rights, lest the economy and the polity be torn asunder by class warfare -- particularly in the wake of the Pullman Railroad Strike of 1894, after which the laissez-faire / company town model began to fall under suspicion.
Trump hasn't explicitly called for greater labor rights, but he's still in his campaign for the Republican nomination. All the calls for bringing jobs back from off-shored companies, restoring our manufacturing base, kicking out the illegal immigrant job-stealers here, and investing government revenues into re-building our infrastructure instead of playing video games in the Middle East -- it couldn't be a clearer signal to blue-collar workers, and a warning to business owners who don't mind ruining the entire society just to add a few extra zeros to their net worth.
Fourth, McKinley's path toward the Republican presidential nomination in 1896 sounds all too familiar:
[His adviser Mark Hanna], on McKinley's behalf, met with the eastern Republican political bosses, such as Senators Thomas Platt of New York and Matthew Quay of Pennsylvania, who were willing to guarantee McKinley's nomination in exchange for promises regarding patronage and offices. McKinley, however, was determined to obtain the nomination without making deals, and Hanna accepted that decision...
The bosses still hoped to deny McKinley a first-ballot majority at the convention by boosting support for local favorite son candidates such as Quay, New York Governor (and former vice president) Levi P. Morton, and Illinois Senator Shelby Cullom. [Compare to the regional favorite sons of the 2016 season -- Rubio, Kasich, Walker, Graham, etc.] ... Wyoming Senator Francis Warren wrote, "The politicians are making a hard fight against him, but if the masses could speak, McKinley is the choice of at least 75% of the entire [body of] Republican voters in the Union".
The main difference comes with the general election, where McKinley faced off against the populist Democrat William Jennings Bryan. This time around, it's almost certain that the populist (Sanders) will not get the nomination, and Trump will be running against a reviled and corrupt Establishment figurehead (Clinton). McKinley acknowledged that Bryan was the better public speaker and did not tour the country drawing the massive rallies that Trump does. Again, personality-wise Trump is closer to Teddy Roosevelt.
Fifth, however, McKinley like Trump was a huge hit among urban and industrialized regions, and less so in sparsely populated regions, particularly out West. Only now the urban industrial core has become the Rust Belt, so Trump is promising to make America great "again," while McKinley was promising to get the fledgling American industry out of the nest and flying on its own for the first time.
It's also unclear how similar his pick for running mate will be. In his first campaign, McKinley chose Garret Hobart, a close contemporary who had become rich as a corporate lawyer and successfully run for office in New Jersey, and who like many in the NYC area had been a lifelong Democrat -- before marrying into a Republican family. He largely agreed with McKinley's policies and was chosen to provide some geographic balance (McKinley hailing from Ohio) and some reassurance to East Coast business interests. As VP, he took a no-nonsense, activist role as the President of the Senate.
(Teddy Roosevelt was chosen as running mate during McKinley's re-election campaign, after Hobart had died of a heart attack in office.)
Trump already has the NYC businessman angle locked down (and by extension the Northeast in general), and the only swing states that boast -- or used to boast -- urban industrial centers are all in the old Northwest Territory, or the eastern Midwest (Wisconsin, Illinois, Indiana, Michigan, and Ohio). Trump could provide balance by choosing a Southerner, but there aren't any swing states to pick up down there. And as in the first Gilded Age, crossing the Mississippi River would be a bridge too far -- then you're getting into solid "values-oriented" cuckservative territory. The Trump movement is a shift of political power back East.
If history is any guide, Trump will go with someone close to the Great Lakes, and more likely from a state that entered the Union early through the Northwest Territory than later on through the Louisiana Purchase. That would still preserve the geographic balance of McKinley and Hobart -- just switching which one was from the Great Lakes and who was from the Mid-Atlantic. He won't be a local favorite son, either, but a low-key independent who may have been a solid Democrat not too long ago.
The only major dissimilarity under a Trump presidency would be the matter of territorial expansion and imperialism, where McKinley was operating when the United States still had plenty of room to expand, and the will to take it. Room to expand has been filled up, and we're mature enough of a polity / empire to not want to keep growing and growing. If anything, we wouldn't mind shedding the deadweight of territories like Guam and Puerto Rico, which we picked up under McKinley during the Spanish-American War. Hell, cut Hawaii loose while we're at it -- by now it's just another generic Pacific Island, not part of America (we see the kind of Presidents it produces).
Jettisoning the territories will happen soon enough, whether under a Republican who appeals to saving administrative costs, or a Democrat who appeals to national self-determination.
Aside from the matter of national expansion (and related to that, warhawk-ism), the Trump realignment is shaping up to be a re-incarnation of the McKinley realignment. Thus Trump is more of a harbinger than a realization of our era's re-born Teddy Roosevelt. Who that will be, we'll just have to wait and see.
 We're praying for Trump's safety, but if it should happen this time around, it would probably be a radical Muslim of recent immigrant background.
 From an 1892 speech:
Under free trade the trader is the master and the producer the slave. Protection is but the law of nature, the law of self-preservation, of self-development, of securing the highest and best destiny of the race of man. [It is said] that protection is immoral.... Why, if protection builds up and elevates 63,000,000 [the U.S. population] of people, the influence of those 63,000,000 of people elevates the rest of the world. We cannot take a step in the pathway of progress without benefiting mankind everywhere. Well, they say, 'Buy where you can buy the cheapest'.... Of course, that applies to labor as to everything else. Let me give you a maxim that is a thousand times better than that, and it is the protection maxim: 'Buy where you can pay the easiest.' And that spot of earth is where labor wins its highest rewards.