cyclists can do a lot of damage too running lights so equal fine seems reasonable, not all fines are set according to social harm "tax evasion ?", dropping litter attracts £50 !
You'll be pleased to know that the fine for skipping lights is indeed the same for motorists and cyclists then
http://www.cyclelaw.co.uk/cycling-offences-jumping-red-lights
The punishment is also applicable to cyclists who, if caught, can be issued a Fixed Penalty Notice of £30.
Despite the potentially serious consequences, only 4% of all pedestrian injuries as a result of red light jumping have been caused by cyclists with the remaining 96% involving motor vehicles
The main issue is obviously enforcement. No police = no fines. Some areas of course have red light cameras which can't catch cyclists, which comes back to your point about registration plates.
On a bicycle, where could you put a license plate that:
i.) Would be visible from an appropriate distance
ii.) Would not get covered in dirt flicked up from the road
iii.) Would not obscure bike lights/other accessories
I take part in cycle events where you're given a number to attach to yourself and your bike. The most visible of those is the one you're supposed to pin to the back of your jersey, but that's not remotely practical/realistic in day-to-day cases. Sometimes you're given a sticker that wraps around the seat post, but that just flaps about, and if you used something sturdier (as per pro races) you get absolutely no rear visibility of the number (so useless for fixed cameras). On the front, the most common solution is a sheet of paper that you cable tie to the bars. These are also a pain due to the cabling around the bars, as well as light/camera brackets, bell fixings, etc. Then you've got the issue of lighting - car number plates obviously have to be illuminated. Doesn't seem very practical on a bike.
edit: I appreciate you've linked to how Singapore are implementing it on e-Scooters, but the geometry of a scooter is quite different to that of a bicycle, and it doesn't look like their solution would work on a bike.
Then we come on to two further practical points:
i.) How high do you think uptake of this scheme would be? You can attach any kind of fine to non-compliance that you like, some people still won't do it, and what are the chances of them being caught?
ii.) How many prosecutions do you think would reasonably benefit from this? In all of the cases that I'm aware of where a cyclist has caused significant harm, they've been tracked down pretty quickly (normally because they've come off equally badly to whatever they've hit)