Based on this: Equal in distribution but unequal almost everywhere?
My answer to questions like those is usually
Flip a fair coin. $1_H$ and $1_T$ have the same distribution but are never equal.
I didn't consider that independence could make a difference. Okay so if $X$ and $Y$ are identically distributed, independent in a finite sample space, what's a simple example to say that they aren't a.s. equal? What about for countably infinite sample space? What about uncountable sample space but RVs still discrete?
I have a feeling I'm missing some simple example to do with flipping a coin or rolling a die.