I had just assumed that it was very rare for masters to want to take their beastkin outside of the country, so they didn’t think to check. Now that Master and Mel ran off they might start being more careful though.
Also, this isn't exactly a police state. It'd be pretty difficult to enforce any kind of strict restrictions on freedom of movement in a turn of the century setting.
You'd be surprised. Remarkably harsh travel controls can be implemented already with Medieval technology, as demonstrated by eg. the Tokugawa bakufu and some of the Chinese dynasties (starting with the Qin) - it's just a matter of state organisation and seeing a reason to invest the required effort. And Forsyth's The Day of the Jackal has a pretty impressive study of the kind of dragnet that can be created with basically just telephones and card indexes; the story's set in the Sixties but the tools employed by the authorities date to the "Long Nineteenth Century".
All true, but for all the apparent new "evidence" we've been given about the legal status of beastkin, we still don't know very much about the specific laws and practices of this storyworld and how they do or don't fit together.
Masters traveling with beastkin within the country is completely normal--we've seen Stella and Master both do it.
There's some kind of status regarding Mel's collar and an ID number, but few of the other beastkin wear anything resembling Mel's bondage/slave/pet collar, which she wore before being taken in by Master.
There's an illegal beastkin slave trade, the specific reasons for which (sex? perverse collections? ransom? pelts?) being left to our imaginations, and the authorities in general seem to be on the lookout for the welfare of beastkin.
Romantic relationships between humans and beastkin are frowned upon, but whether they're actually illegal, and whether the objections are to interspecies miscegenation, to (possibly) unavoidable pedophilia with the never-fully-maturing beastkin, or to inequality of social status, or some combination thereof, is never made clear.
The ending implies that there's some sort of "Transporting Beastkin Across International Borders for Immoral Purposes" issue, but its nature and extent is, once again, left to our imaginations.
Ah, well, like Marie and Collette, I just want the people, etc. I care about to be happy together.