Depending! I don't think that they would like permanent marks like logos - seeing the Weyland Corp Logo on David's fingertip makes me wanna cry.
So I don't think that they would like marks of possession... I mean, they would surely like to be able to carry a hickey around for a few days, showing others how passionate their s/o is about them, but nothing that stays and says they are a product.
(That just made me think of Andy and Thia/Tessa. While David is probably able to slice the Logo off...what are they to do? It's on their fucking eyeballs. This is so awful QWQ)
But I believe that some of them, mainly David, Walter, Bishop and Thia would like tattoos! Be it permanent or just drawn on them for fun. They like the meaning behind it, and being adorned with art. Walter with some plant tattoos? David with some nice LoA motives or stars? Bishop got convinced to have a marine tattoo, being even more included by the Sulaco crew! Thia just goes ham - maybe she gets blooded by Dek? She would wear that with pride.
And with marking their s/o it would also depend on what exactly you mean. I think Synths would enjoy giving their humans hickeys and scratchmarks if their humans want it :3 (David would like to do it more than the others, though, for obvious reasons, I think he'd also be very entranced by seeing the marks fade, quickly creating new ones <3)