I suspect it's more common with smaller aircraft where the margins are slimmer with the usual configuration of 2 chaps and full fuel. I would imagine that almost every training flight in C152s where I trained takes off overweight.
This is not acceptable, ever.
As a flight instructor, I have no tolerance for an operation out of safe limits, or operating outside the aircraft limitations. That includes the weight and balance. A student must complete a weight and balance calculation before every flight, and it is NEVER acceptable to operate, or allow a student to operate overweight or with the center of gravity outside limits.
Particularly in the instructing arena, it's a sacred responsibility on the part of the instructor to provide an example. A student should never be given the impression, not even implied,that operation that is unsafe or illegal is in any way acceptable. It's not.
If a student wants to fly in a Cessna 152, then the numbers must work out...otherwise the student will be moving to a 172. Even if the numbers work out but performance is compromised to the point of safety (very likely at many of the airfields where I've instructed...which have 10,000' density altitude or more in the summer), then we're moving to the bigger airplane.
Overweight operation isn't acceptable in any other kind of flying operation, either. I often find that weight must be reduced for performance reasons, rather than the actual weight limits, before the weight limits are ever reached.