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?

I like it. 

When I need to move fast to e.g., catch a plane, I take a taxi. 

Taxi drivers need to have a license, so they pay back some money to the city for utilizing the bus lane.

Building @kiwi | https://macbudkowski.com/

The point of bus lines is that it incentivizes people to have a viable alternative to riding/driving a car. If taxis makes this less viable it will end up leading to the same problem as if everyone rides a car. Public transport demand has to be induced

I am a public transport maxi and never seen it causing any problems in Warsaw (can't say for other cities). On crowded streets buses go every 2-4 minutes, even 60 taxis per hour wouldn't really impact the traffic.

Some people never take a public transport, so at least they take a taxi instead of blocking the roads while looking for a parking spot.