Why are salamanders illegal?

Because they burst into flame! 90% of wizard dwelling fires are caused by salamanders!

(in reality probably the law banning them as pets to protect them didn't include axolotls because the legislature didn't know they existed)

They're either an invasive species, and therefore should not be introduced to the area (and you know that many pets will be introduced once the novelty wears off). Or they're native to the area, and should be left alone because they're endangered or otherwise threatened.

Those are just two reasons, but I'd bet they cover a lot of cases.

Often Axolotls have been "grandfathered" into the legal exotic pet trade, and salamanders have not and they tend to be considered separate species, even though biologically it's a very blurry line. Also, it often happens in areas where there is a local wild salamander population that is being protected from poaching.

You likely don't have wild axolotls nearby so if a pet escapes it'll just die and not affect the ecosystem. OTOH, an escaped salamander might thrive and displace wild salamanders and disrupt the ecosystem. Or carry a disease, or ...

most places ban exotic pets that are able to survive in the local climate to prevent invasive species from outcompeting the local feral cat population.