LOL, it's as if Bethesda gave a big FU to furries.
Argonians aren't missing from the list of people you can marry, so not really. Furry applies to anthropomorphic (human-like in form and/or personality) animals in general. I understand that's confusing. Then there's the fact some sixes are missing for some races and the Bosmer are also totally missing. It's bizarre, and hopefully not for something as silly as running out of room for audio files on one of the three versions of the game.
It isn't only something that would annoy furries though. What if someone wants their Khajiit to marry within his/her race? Someone isn't necessarily a fan of anthropomorphic animals in general (which is what a furry is, there is no rule on just how much they like them) all because they like Khajiit or Argonians.
Well, you can't marry him (unfortunately, and from what I've seen while wearing the amulet), but you can have a Khajiit companion follow you around. His name's Kharjo and he's travelling with one of the caravans. You'll have to complete his quest first.
Spoiler J'zargo, a male Khajiit student in the College of Winterhold, can also be a companion.