Here is how I read the effect of the star card.
Say you have a base stat that you have raised to 20 naturally. I am not going to consider other methods right now because I don't want to delve into precedent of magical effects and their interactions. I just want to address what I perceive the effects of the star card to be.
Increase one of your Ability Scores by 2. The score can exceed 20 but can't exceed 24.
There are two statements being made here. First: Increase one of your Ability Scores by 2. This statement is very straight forward you increase an ability score by two.
Ok, so how does this interact with the rule in the PHB that says...
You can't increase an ability score above 20.
This is handled by the second sentence in the effect. The score can exceed 20 but can't exceed 24. By that effect your score now is able to be raised to 22.
The fact that this effect is broken into two sentences that individually have a stand alone effect tells me that the effects are in fact independent of each other. If the stat increase was directly tied to the stat cap increase I believe there would have been one major difference. The effects would have been joined by a conjunction for example the text of the Ioun Stone of Leadership states:
Your Charisma Score increases by 2, to a maximum of 20.
This basically says the stone functions exactly like an ASI with regards to Charisma.
Similarly, the Book of Exalted Deeds says:
Wisdom score increases by 2, to a maximum of 24.
This means the book will work similar to the rules for ASI's except it can increase wisdom by 2 up to 24 instead of 20.
Finally, all of the Tomes of X use the phrase:
Your X score increases by 2, as does your maximum for that score.
This is in my opinion the most powerful maximum increasing item type, since it does not limit the maximum possible attainable maximum. So in theory a character that hits 20 and has an ability that makes them long lived they could keep using it every 100 years until the hit the 30 hard cap as indicated here by JC https://twitter.com/JeremyECrawford/status/973771989499310080
The star card specifically phrases it differently. It has two independent clauses that address two different aspects of your character. The argument can be made that it is not intended to be read this way because of this tweet by JC https://twitter.com/JeremyECrawford/status/991012429466251270
It is a strong argument and I imagine it would be up to individual DM's to arbitrate which way to read this but I still see the individual sentences having related but independent effects.
As to what would happen if you draw multiple star cards it is simple.
Draw 1: Both effects take place score goes up to 22
Draw 2: First effect takes place raises score to 24 second effect is irrelevant since Draw 1 already raised max to 24.
Draw 3: Nothing happens score is already maxed at 24.