There are 50 states in the USA.
When does one use Bundesstaat, Bundesrepublik and Bundesland? If I understand correctly Nord-Rhein-Westphallia a state is a Bundesland and Germany is a Bundesrepublik. But the states of Texas and Florida are both Bundesstaaten.
I don’t know why, but for different countries we use different terms, probably somewhat related to what term the country uses for their own parts.
Beside “der Bundesstaat” and “das Bundesland” there are also “der Kanton”, “die Provinz” und “die Präfektur” for other countries. There are probably much more, but those are the ones which came to mind immediately to me.
Die “Bundesrepublik” is a bit different since it depends on the political/legal system of the country. “Republik” is like republic in English, and if i remember correctly, the “Bundes” comes from it being a “Bund” (union) of different parts, connecting to one republic. And as you said, it refers to the country itself and not their parts.