Battery difference between models?

Hello!

I bought my XS Max in March 2019 and have been using it with the original battery ever since. Somewhere in 2020, I went to an Apple Store to replace my battery but they told me the stores in my country didn't offer services for the model I'm using (A2102), even if I pay the full price.

They also told me that each model used different parts (battery/screen etc), so they wouldn't be able to offer me the battery replacement for A2102 model, they would only offer support for A2101 and A1921 models. Is this true, or was I misinformed? Would any battery I buy from iFixIt work with my A2102 phone or do I really need to search for batteries for the exact model I have?

Sorry if my question doesn't make sense, English is not my native language.