I think everyone would agree that it is possible to be neither or both, but can you be one without the other?
The definition of "partisan" I like is this one from Merriam-Webster:
a firm adherent to a party, faction, cause, or person; especially : one exhibiting blind, prejudiced, and unreasoning allegianceKey in to the part about "unreasoning allegiance".
I think one can be a supporter of a political party, without being partisan - that probably takes little explanation.
I also think one can be a partisan of a political party without supporting it (that is, unreasoning allegiance). In this sense, you may publicly support the political party despite the fact that it seems not to follow your political belief structure.
Could someone realize, for example, that all the time they were a partisan of one party, they actually (perhaps) belonged as a supporter* of another?
* Note supporter vs. partisan.