Two shaver sockets on same circuit with extractor fan
the extractor fan lasted about 3 weeks before buzzing and then completely failing (but still buzzing)


Perhaps a bearing seized up, It is probably a small extractor fan and is easily replaced at low cost. You can do it yourself if you follow basic electrical safety procedures. 

I replaced a noisy (but still working) bathroom extractor fan with another of the same make and model which also started to become noisy after a few weeks. I then replaced it with another make (Manrose Gold - "Long lasting ball bearing motor".) which is still working fine and does its job quietly and reliably. 

I conclude there are good makes and bad makes of bathroom extractor fan. It may depend on which factory in china they use and who does the quality assurance. I suggest you don't buy the cheapest you can find.


  did the two shaver sockets destroy the extractor fan?


It was probably a coincidence.


  Does the extractor need to be on a separate circuit?


No.

The total expected load on a circuit should be less than the maximum allowed for the wire cross-sectional area and less than the rating of the circuit breaker for that circuit.

Charging a toothbrush probably takes a very very small current, maybe 0.01 Amps. This should not be a problem.

A 20W extractor fan on a 230V lighting circuit needs about 0.1 Amps. This also is a pretty small load and not worth installing a separate circuit for.

Your lighting circuit is probably rated for 5 Amps total load.