How does British Airways permit a 23 kg bag in hand baggage?
Easyjet has no firm weight limit and advertises this fact. From the website


  There’s no weight restriction for cabin baggage, as long as you can place and retrieve the cabin bag safely in the overhead lockers without assistance.


They also use vague language like “within reason” or “lift comfortably” but most people probably won't have 20 kg with them. Given the limited size of cabin luggage, you would need to deliberately carry dense, heavy stuff to reach that weight (books maybe? wine bottles and other liquids are off-limits and if you don't want to check in any bag you still need space for your other things).

And then, comes this (from the BA website you mentioned):


  Occasionally we may not have space to stow all hand baggage safely on board and you may be asked to check in some or all of your hand baggage at the boarding gate.


Bottom line: Airlines don't need to rely on a prespecified weight limit at all. They rely on the size restrictions and general common sense to keep things manageable and if it somehow does not work out, they can always gate check a few bags.