Well right now that is changing dramatically. The "alt-right" is drastically changing what it means to be a Republican. Meanwhile Bernie supporters and an organization called the "Justice Democrats" are drastically redefining what it means to be Democrat.
This is actually one of the primary reasons Trump won. Everyone in the USA is sick of the status quo, e.g. "the way things are". Gallup polls are showing faith in public institutions are at all-time lows.
No one thought Trump had a chance, all through the primaries. More relevantly, no Republicans thought Trump had a chance.
Bernie supporters knew Bernie was the obvious best choice to defeat a Republican candidate. Hillary squashed the Democrats' chance of winning this past election.
Because Trump represents the new Right Wing in politics in the USA. Bernie represents the new Left Wing.

This is the era of Millennials coming of age. Baby Boomers are dying off in droves.
The details of your question - what is a Democrat and what is a Republican in this day and age - are being hashed out in the Courts and in Congress and we speak.