This comes up from time to time so if you do a search you'll find more detail.
In the UK certainly, ATC has a variety of different categories of emergency that can be declared - none is called mayday or PAN. ATC decide, either on the basis of experience and professional judgement or procedures, what category to declare. ATC can declare an emergency even if the pilot does not make a mayday or PAN call. If a pilot does make an emergency call, ATC will certainly declare some level of emergency.