Great question!

The word comes from the Latin Exonym, but that does come from Gut-þiuda which means "Gothic people," at least as far as modern historians think based on the Pietrosa Treasure. So yes, they did call themselves "Gothic people." But this was in no way their primary identifier, as they called themselves by the names of their smaller identities like Terfingi, Greuthungi, Amali, etc. etc.

But identity based on modern archaeological or etymological groupings isn't at all relevant to how these peoples saw themselves. A Goth and a Vandal wouldn't think of each other as similar, they don't know they share a language family or similar art styles any more than the average person today does. On a basic level familial ties were the most important to these Germanic groups. This is best explained by Walter Goffart in his Barbarian Tides. We see this most obviously in Jordanes' Gothic history where his work revolves around either fabricating or exaggerating the Amali family line and its ties with other Barbarian leaders.

I hope this answers your question. I would actually strongly recommend the aforementioned Barbarian Tides for understanding these peoples better.