This is only an issue if you want it to be. Technically most droids are also property, making them slaves as well. Nobody has any issue with droids roaming around.
In all likelihood, your Wookiee PC would have to worry about added scrutiny on the part of Imperial officials or xenophobic administrators. A Star Wars equivalent of a conceal and carry is probably out of the question, as would the IPKC bounty hunter license.
Actually a wookiee could get an IKPC point in fact Snoova had one in this era.For me it depends a wookiee without certain obligations is criminal or bounty have a document that says he is his owner.If he doesn't than Ya he would be considered a slave.Of course I also put restrictions on establishments for droids,aliens and even humans.