The reason for removing the +ve first is that the whole car body is -ve. Removing the +ve side of the battery can ensure that even if the dangling +ve wire touch any part of the car body accidentally, there is no electrical short. It is just logical step as extra precaution though not mandatory.

In olden day, there are few car model that has +ve body (+ve side of battery connected to car body). Not sure if there is still car having that configutration nowadays.

The problem with modern cars is that some setting are lost when battery is removed. When battery is weak, you may not able to start the car but there is still power left in battery to retain the setting. Therefore, mechanic will jump a cable to external battery to keep the memory setting before removing the battery. Put new battery and only then remove the temporary external battery. Some just jump start the car with external battery or with other car, By leaving the engine on, the memory setting is retained even when the old battery is removed because alternator is taking chage to power the car when engine is running.