In the UK (which the article seems to be about, although it keeps talking about other cities too), it depends on the specific road markings. If it's a solid white line between the bike line and the road, absolutely no vehicles are allowed into the cycle line except bikes. If it's a dashed line, they can enter it as long as it's not being used by cyclists.

In general though, special affordances for parking and dropping off disabled passengers is only given to those with a blue badge. AFAIK in general taxis carrying disabled passengers without a blue badge have no more rights than any other vehicle. Only companies that specifically care for disabled passengers can get their own company blue-badge rather than relying on their passengers having one.

Right, but in general, what is a car driver supposed to do when he needs to stop and unload (anything) in a place where there isn't a special provision for that? Here in Slovakia they can stop in the car lane, blocking it, as long as that's not explicitly forbidden by a sign there.

Because we also have streets with a car lane and a bike lane with a solid line where it's forbidden for any other vehicle to enter or stop there, so of course I had to explain to a few drivers what are the rules they should alrady know and follow.

It depends on the road markings and signage. In general double yellow lines mean no waiting or parking, but there is generally a special exemption for loading/unloading or letting passengers in or out. However, there is frequently additional signage that will restrict hours when loading is permissible, or state no loading.

Double yellow lines (waiting / parking) are different to bike lines however. If a bike line is separated from the road by a solid white lines, motorised vehicles are forbidden from entering at all. If you want to load/unload or drop off passengers, you need to do that somewhere else. If the highways agency considered it safe for vehicles to block the cycle route in that place, they'd use dashed lines instead.