Yes, Lincoln's policies were ideologically equivalent to the modern GOP's
No, Lincoln's policies were not consistent to the modern GOP's
There were all along since the Civil War, more racist democrats than republicans. Racism still exists in both parties, but even today. More democrats are racist than republicans.
More republicans as a percentage voted for the 1964 Civil Rights act than democrats.
Too many democrats believe in quota systems even today, which means they think blacks are inferior, and need help. That they can't compete equally.
Think about it. Prejudice is not racism. Racism has to do with believing another race is inferior.
"No religion is true, but some religion, any religion, is politically necessary. Law and morality are insufficient for the large majority of men. Obedience to the law and to the morals are insufficient for making men happy. […]Law and morality are therefore in need of being supplemented by divine rewards and punishments."
In '64, around 80% of the republicans voted YES for the civil rights act. Only around 60% of the democrats did.
Just one reason MLK would align himself with many republican. He has stated he normally voted democrat, but that was before the Civil rights act, and just voting "mostly for" should tell us he was likely not partisan. Still, she claims he had registered as a republican. He may have previously been registered as a democrat. Without digging up the archives, we really don't know, and I haven't seen such work done.