I hear 別に used as "not in particular", did this usage come about just directly from the meaning of "separately"? So when used, it means that someone has a separate interest, like, etc. and hence they don't particularly have the interest, or share the like?
No, this negative expression means "Not really", "I'm not interested", "I don't care", etc.
I think this 「別に…」 is the shortened form of "別に言いたいことはない" or "別にそういうわけではない", and 別に here roughly corresponds to particularly.
Something positive, like "これとは別のものが好きです" or "私はこれとは別に、言いたいことがあります" is not what the speaker want to say.