Yes, normally, but in the gamified fanfiction that is her Origin, she is Tav and hence desirable and desired by any and all. Including the bad boy beau bloodsucker, who squeezes her name into basically every line during their modified tête-à-tête in the woods, so that it really does sounds like exceptionally clunky fanfiction, dreamed up by the horny-teen-soldier who desperately needs romance in her life.
Whenever I feel a little disgruntled that they didn't even bother to change the approval for blood drinking during sex for Gale, the way that scene was handled with Karlach reminds me that there are worse things than neglect.
Ah, yes! He indeed says her name in every sentence. He keeps repeating it in the epilogue too. So much cringe!
The contrast between his canon behaviour towards her when you play as Tav and when you play as Karlach is just hilarious.