In multiple European cities now I see taxis being allowed to use the bus lane (while of course regular cars are not).

What's the policy justification for this?

in Lisbon it’s only official city taxis that are allowed, uber and bolt aren’t

Founder @basedstudio.eth. Product Maxi & Idea Guy. Forever Intern. Pushing Higher. Network States, d/acc ↑

Which implies that it's because official city taxis pay some kind of tax, and so this is an obfuscated way of paying for priority. So... why not just add a pay-for-priority-passes feature directly, and let people with their own cars also benefit from it if they choose to pay the same tax?

Yes I think the only thing stopping from implementing that is the bureaucracy required (registry?) and extra work on the visual of the car (proof that you have it). 

I think that’s why people with their own cars can’t benefit from it on per ride basis, they’ll need something visible that they can be turned on?

Because the feature of a relatively-jam-free lane reduces as more people join that system. The priority lane's a physically scarce commodity (2 cars cannot occupy the same physical space), thus is unfortunately susceptible to the tragedy of the commons (even of the commons is artificial).