Does this have the same meaning in English as someone you're in a serious romantic relationship with, but don't wish to specify the gender of?
No. It may mean a serious romantic relationship but you think boyfriend is inadequate. If the partner is female she is "partnerka". It can also mean business partner or dancing partner etc. But men. Only treat sb like a partner is for men and women
I think that in some contexts it could mean "partner" as a general term but rather not in romantic relationship.
Yeah, it's possible. You can also use it in a heterosexual relationship to refer to your boyfriend, although that sounds... formal? And then, it can be a businesspartner or someone like that.