This answer is extremely brief for such a broad question, but I'm sure others can expand on it more. Forgive me if something isn't clear.

Technically, the current Republican Party is the same party that began with Lincoln. However, it's much more complicated than that since there has been a number of party realignments since the Republican Party was founded in 1854. It was founded as a mostly Northern anti-slavery party and for many decades was considered the most Progressive of the parties. Democrats, on the other hand, were mostly libertarian and fought many Republican regulatory policies.

The Republican Party as we know it today began in the early 20th century with the emergence of the Old Right faction within the party. The Old Right favored limited government and free market economics and were among the most vocal opponents of the New Deal. Meanwhile, FDR's New Deal Coalition brought together urban populations affected harshly by the Great Depression, taking away huge numbers of people who previously voted Republican. However, both parties were still a mix of what would be considered today to be liberal and conservative views on government. The Old Right disappeared with their losses under the New Deal government, but the Republican Party remained in favor of more limited government.

The last major shift to form the modern Republican Party was the Civil Rights Movement. As the Democratic Party became more Progressive, more and more "values voters" flocked to the Republicans, who were not proposing radical social changes. As social issues became more prominent, the Republican base became more well-defined and the South became a Republican-dominated region because of their desire to maintain their way of life.

So, in short, yes, it is technically the same party as under Lincoln, but it is also very different. As different issues have come up over the last 150 years, Republicans have taken sides and the party has changed drastically because of it. Republicans were never "progressive" in the modern sense, but they supported limited government involvement for many years. It was only after the New Deal coalition went farther left than they were willing to go that they became the "conservative" party, still pushing for limited government. So, whether it's the same party is a matter of perspective and opinion. We don't know how Lincoln would have felt about tariffs or abortion or civil rights because they were not the issues of the day. Party realignment is a natural part of politics, so I would say yes, it is still the same party.

Edit: Fixed a typo