There are some similarities with the four political parties listed in your question. I will highlight some of the ways each political party is similar.
The Federalist Party tended to be supported by businesses. They advocated for a stable financial system and supported the concept of having a national bank. The Republican Party today also has a lot of support from businesses and from the banking industry.
The Federalist Party believed in a looser view of the constitution. They believed the federal government should have a lot of power. Many members of the Democratic Party share these views today.
The Democratic-Republican Party wanted to limit the power of the federal government. They supported having low taxes and a smaller federal government. They believed the states should have a lot of power. They also supported a strict view of the constitution. The Republican Party today shares similar views.