Let me try to answer this as best as I can while my best sources are packed away in boxes...

First, while there are similarities between the two time periods, there are key differences as well. There was no real second generation of Federalists after Alexander Hamilton; by the time of the War of 1812 many of the Federalists had passed away or were no longer in government. Jeffersonians had no real political opposition and even adopted some of the Federalist platforms, such as more government involvement in economic growth. On the other hand, the Democratic Party did find "hideaways" in America in the immediate post-Civil War era

Second, after the Civil War, the nation remained very divided -- arguably even for another century -- between the old Union and the old Confederacy. Southerners had walked out of the Democratic Party convention in 1860 over the issue of slavery and nominated John Breckinridge, who eventually ran in a four-way presidential election in 1860. The nation thus had a northern and a southern Democratic party.

Reconstruction largely did not change attitudes of Southerners, many of whom vehemently opposed the pro-civil rights platform of the Republican party during the 1860s and 1870s. These southerners voted in larger numbers towards the end of the 1870s and 1880s, putting southern Democrats back in power. In fact, after the Rutherford B. Hayes election in 1876, a Republican did not win an electoral vote from a southern (I define "southern" loosely; strictly I mean AR, LA, MS, AL, GA, SC) state until 1964. Incidentally they all went Republican in that year, in no small part due to the party's opposition to the Civil Rights Act.

During Reconstruction, northern Republicans still harboured racist and prejudicial sentiment towards blacks, and thus were not enthusiastic about challenging the party's stance on civil rights. Moreover, the post-Civil War die of the American South had more or less been cast by 1877, as the Supreme Court increasingly ruled against civil rights legislation and violence against blacks returned. By 1896, segregation was the law, thanks to Plessy v. Ferguson. The biggest political issues of the time were no longer slavery, but the agrarian-industrial divide between the Republican-dominated business interests of the east and farmers further west, and progressive reforms in urban areas

TL;DR Southern Dems still had intense support in the South after the Civil War, and little opposition from northern Dems

Hope this helps!

Source: Hershey, Marjorie Randon., and John H. Aldrich. Party Politics in America. Boston: Pearson, 2013. Print.