Is it me or is this a weird question to a freshman? I mean 'degree'. Isn't 'subject' or 'study' better?
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Not really. In a British university, it would be more common than "subject/study". I was going to say it would be perfectly interchangeable, but actually, it would be a bit weird asking someone about their study or their subject (presuming you would name the subject though, to make it a little less weird).
I don't know if it's because of differences in culture, but here, you generally have your degree set out from the start of university. You got to study for a degree in whatever subject you've chosen, so "how's your degree going" sounds perfectly normal. It's a bit like asking "how's school going?"
"How's study going?" on the other hand, sounds more to me like "how's your homework going?" - it sounds like a specific thing, rather than a more general one.