It's similar, if it's domestic SMS. (In fact, a lot of plans no longer offer "flat rate SMS" rather they are built into now ubiquitous "unlimited" plans.

Other than some carriers offering extra international SMS options, it's certainly not is a standard.

I am not very sure about how it really works in Europe, but I have feeling that EU countries have more comprehensive arrangements for calling/texting/roaming across their borders, or at least I wouldn't be too surprising even if so.

Yes, in Germany we do have "EU flatrate", a set of "favorite countries" or a big package like 500 SMS for foreign numbers.

The option depends per operator.