Yeah it's the latter. The US does not have party membership the way that, say, the UK does. In many states, it's open primary. In Colorado, for instance, I get mailed Democratic and Republican primary ballots and can vote by mailing in either one. I think you get neither counted if you mail in both, but I have no idea; I've never tried it.
The last time anyone tried to poison a presidential election by promoting a weaker candidate on the other side in the US, it was the Democrats boosting Trump in 2016. It did not work out.
For an alternate example, in Illinois you choose one at primary election time and only get that one. This year the options are Democrat, Republican, Libertarian, and Non-Partisan (which means only the referendums, not the elections).
Or, in effect, are you just required to claim either that you're more of a cat person, or that you're more of a dog person?