What do we think happened to cause the BBU / WRU multiverse to diverge in such a way as to allow these things to exist when chattel slavery is illegal in every country in our timeline?
I would open this up to the general BBU public, because indeed those writers with bigger worldbuilding sometimes have established that.
The depth of this worldbuliding is absolutely liberal of course and not necessary at all to be a BBU writer; after all vampire fiction also exists even though feeding on humans is illegal in every country in our timeline, and not everyone cares to elaborate on the biology and politics and history of their fiction. ;)
BBU is a dystopian society that is similar to our world but it's not it.
And just in another response to how the question is phrased, slavery being illegal does not prevent it from existing in the real world either - human trafficking, forced prostitution, forced marriage, taking away workers passports etc do sadly exist.
That said, back to the fictional world of the BBU, I have heard and seen very different takes, that I encourage the followers of this blog to expand on.
it is in fact not seen as slavery, as the future pets sign themselves over voluntarily; it is more of a lifelong servantship that they are paid for handsomely in advance.
Turning people into pets against their will is illegal in pretty much every flavor of the BBU I have seen.
in many branches of the universe, the legality of the pet trade is challenged by other countries and jurisdiction varies
There are also branches of the universe where pet rights exist and a humane treatment of pets is actually enforced (success varies);
Again: BBU is a dystopia! It's a ultra-capitalist society, where companies such as WRU can go pretty much unchecked, corruption is everywhere, and many rules are enforced brutally and without regard to human rights. (How similar from actual real world developments that is, is left for the reader to decide)
Often in the individual it follows up on historical (or current) events where economic stability crumbles and signing up is seen as a chance to save one's family from poverty; or in times with cruel prison sentences it is an opportunity to escape from guilt or punishment. Some BBUs are set in a time like ours, some in a future not very far away from it.