There's no need for either of them to take each other's names.
Seconded.
Of course there's no need, and it's physically impossible for her to anyway.
Right... because names are physical...
Actually, this may apply to my point: Chakotay's equivalent for a last name is tattoed on his forehead.
Now according to the claimed need to assess a family merge in marriage by way of sharing a name, Seven should get a tattoo. Except it's an ethnic thing. Therefore she doesn't need to. (Unless she was visiting the tribe in the jungle and they wanted to adopt her... and she wanted to be adopted. If they could work around the ocular implant.

Hold on, I'm running out of ifs.)
Nor does Chakotay need to use a name from her old Borg or human identity. What might matter is what last name they use, and neither of them uses any.
Pointless debate anyway.
The only people for whom the name choice might be relevant are their hypothetical children.
Oh, and, by the way, I think Chakotay did the tattoo (officially to honor his father) so that he could dispense with most other Native American traditions. Religion set aside, he's a modern man, not a traditionalist, and certainly not a polygamist.
And I believe "squaw" is a derogatory, racist term. Better dispense with it. The use of it in a point about polygamy underlines how antiquated that idea is. In Native American societies, if polygamy was practiced, it was about survival. For all women to be provided for if there were less men, or proper providers, than women.
In a hypothetical generational ship with less men than women, one would rather consider polyandry...