My question stems from this one: Does the Star card from the Deck of Many Things increase your ability score above 20?
How exactly does increasing an ability score above 20, by any means, work at all?
Relevant quotes:
Ability Score Increases: Some of these features allow you to increase your ability scores, either increasing two scores by 1 each or increasing one score by 2. You can't increase an ability score above 20. -PHB pg. 15
Manual of Gainful Exercise: …your Strength score increases by 2, as does your maximum for that score.
Say you have a Strength score of 18, and use a Manual of Gainful Exercise to increase it to 20, can you still not use an ASI to increase it to 22, despite the "maximum" now being 22? In fact, is 20 even the maximum for your ability score when using items; the quote seems to only apply to Class Features.
Best Answer
RAW
From strict reading of the ASI, you shouldn't be able to, since it does not say "You can't increase an ability score above your current cap" or something on these lines, it just says "You can't increase [...] above 20".
Rules as (probably) Intended
Now, is it intended? Probably not.
In the PHB, the only way to go over your cap is through Barbarian's Primal Champion feature, which is at 20th level and you are not getting any more ASIs after that.
So one possible reading on this is that it's worded as it is for simplicity, since they were not considering the items on DMG.
The problem is that, from my experience, the DMG is written (more) vaguely (than the PHB), which might even be intentional since it's supposed to be read by the DM and he can decide whatever he wants. This means it's probably implied that increasing the maximum score by 2 (to 22) allows you to increase your stat score to 22 by any mean, including ASI.
A further indication that you should be able to do it is that every mean of increasing your stats says something on the lines of "up to 20" (AFAIK), instead of "up to your cap". This means any feature increasing your cap doesn't do anything, since everything is hard capped in its own text anyway (Ioun Stone as another example besides ASI).
For clarity, what I mean here is that there would be no reason to word it as "your maximum for that score increases by 2" instead of "up to any value" or "not hard capped at any value" or anything like this if it was not supposed to influence any other item or feature.
About your last question
The quote is a general rule. This shouldn't matter though, because as far as I know, the items explicitly state their cap (Deck of Many Things' star card caps at 24, Ioun Stones cap at 20, etc).