That's not true.

We poll the hell out of our populace and ask, "What do you identify as?" That's how we know how people identify.

> In other countries, people pay membership dues to their political parties and get membership cards. Here, we just declare our allegiance to these private organizations on twitter.

Some states in the US have state-level party organizations which you join and receive a card. Not all.

How does these polls work? What’s the selection process? I’ve never been asked what I politically identify as, and I imagine anyone who asks would be disappointed that I would refuse place myself as something fitting whatever binary/trinary categories they have in mind.

Many states allow you to pick a party affiliation when you register to vote, and some require a declared affiliation before you can take part in primaries.

Generally polling organizations poll a random sample of Americans via phone calls or online polls. There is a lot of process to try to guarantee a representative sample.

I don't think a pollster would be disappointed that you don't have a party affiliation. In fact some of biggest emphasis in election season is on independents or undecided voters.

> In fact some of biggest emphasis in election season is on independents or undecided voters.

Exactly. You can look it up on 538 or wherever you like, but all the polls say roughly the same thing: There's more unaffiliated voters than there is voters for either party. Something like 40-45% unaffiliated, with ~25% identifying as Republican and ~30% identifying as Democrat.

You can pay dues and get a membership card for political parties in US, as well. The difference is that it's not necessary to have access to the primaries in most places.