They share the some of the same constituencies (Mainline protestants, upper income liberals, economic nationalists), but there are a lot, and I mean a lot, of differences. For one, the Gilded Age Republican Party wouldn't have any support from Catholic voters, considering that, y'know, they were virulently anti-Catholic.
1. There are enough similarities that pretty much the entire Gilded Age GOP electorate has gone over to the Democrats (while nearly the entire Democratic constituencies in the South and Mountain West have switched to the GOP). Look all the way down to
county level voting -- it's astounding really.
2. The Gilded Age GOP wasn't as anti-Catholic as the all-Democratic Ku Klux Klan, was it? After all, it looks from the maps that the GOP was very strong even in heavily catholic areas of the North (Rhode Island, Connecticut, Mass., Wisc., Minn.) while the Democratic Party was week in Catholic areas of the South (like the French Catholics around New Orleans and the German Catholics in central Texas).