Yes, trams in the UK have never carried registration numbers, and their drivers were not licensed either as part of the motor vehicle (car) licensing system or via the Traffic Commissioners' PSV licensing system.
I can't think of any tram system where the trams never carried fleet numbers fairly prominently, but not sure if this was a legal requirement.
Prior to the new generation of trams appearing at Blackpool, many tram drivers also held PSV / PCV driving licences, but that was because they had previously been bus drivers, or were employed mainly as bus drivers but drove trams in the summer. (I don't know if that's still the case.)
Likewise, 'first generation' tram conductors did not need a PSV conductor licence (when they were still a thing) although most if not all of Blackpool's did for operational flexibility.
I'm less sure how common 'dual badge' crews were on other 'first generation' tramways.
It's only come in for the most recent generation of trams (possibly when new laws were drafted when the second generation of trams were being planned?) that tram drivers were required to hold a car driving licence first - and I think the few existing tram drivers who didn't were allowed 'grandfather rights' to continue driving - I seem to remember something in enthusiast media some years back when the last Blackpool tram driver who had never had a licence to drive motor vehicles retired (from memory, he'd moved from Sheffield or Leeds when their trams went, rather than go bus driving.)
Everything around trams was and is subject to approval by the board of trade / railway inspectorate / whatever it's called now - this would include driver training and testing.
Trolleybuses are different - they were road vehicles, but not motor vehicles - they carried registration number plates, and trolleybuses were a class on a normal driving licence (issued by local authorities before DVLA was set up in the 70s), but in some respects they were subject to 'light railway' law not road traffic law. Drivers and conductors were licensed by the local authority (or in London by the Metropolitan Police's Public Carriage Office) rather than via the Traffic Commissioners, so again if they had a PSV licence / badge, this was so they could also work buses (some operators ran both and were that flexible. In London, trolleybuses were part of the 'tram and trolleybus' department which was separate from the central bus department until 1950, the terms + conditions were different and the uniform subtly different so at one stage, the two did not mix. As the conversion plan got going, there were a few depots / garages that temporarily operated trolleybuses and buses at the same time, so there was a short time when crews who had nominally become bus crews could work overtime / rest days on trolleybuses.)
I understand that the law / driving licence class for trolleybuses stayed in existence for a decade or two after the UK's last trolleybuses ran, but I think the relevant laws have lapsed - it may have happened in the early 90s when the driving licence classes changed and the old separate PSV licence became class D on the Swansea issued licence. (this is from memory so may be adrift.)
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