Firstly, Pening, it's perfectly normal to have a standard bathroom extractor connected to the lighting circuit. It's a very small load. Also, it's usual for it to be linked to the light switch. When the lights come on, so does the fan.
Officially, any bathroom work is notifiable (it has to go through Building Control.) If an electrician is "Part P" certified, then he can do the work without having to involve Building Control directly (this called "Self Certification")
A handyman is not likely to have done this. At the same time that the fan was fitted, it should all have been tested, and an RCD (earth trip) added to the circuit.
Assuming that a "Part P" electrician changed the consumer unit, then this would include an RCD... so ok there.... AND he would have self-certified.
This same chap issued the EICR. This covers all circuits and fittings (unless stated on the certificate because of restricted access etc)
The fan would have been included in the tests.
All fine there, but it leaves one possible issue. Was the original fan installation ever notified? It seems not. From a safety point of view, your whole installation has passed as satisfactory. That's the main thing. The original notification issue is merely a legal one.
Seeing that this is a rental, to put your mind at rest, ring your local Planning Dept. (District Council) and ask for Building Control. Ask how you can put the notification problem right. I anticipate that they will ask for a "Minor Electrical Works" form be submitted (inevitable fee of course.)
All of the above applies equally to the kitchen fan. Kitchens, as with bathrooms, are considered "special locations".
Assuming the handyman ducted to the outside, then, mechanically, that's fine.