Imo there are two aspects to this. First is type of fish. Second is what you mean by do they 'care', whether you mean can they adapt and lead a healthy life or if you're talking about how comfortable they'll be or how their behavior will be affected.
First, type of fish. Size of fish makes a difference-- ratio of water depth to fish size. Shape of fish makes a difference-- tall bodied or vertical body plan fish (like severums or angelfish) vs more longitudinally oriented fish. And behavior makes a difference-- bottom oriented fish vs. more mid-water to surface oriented fish.
Second the context of 'care'. A substrate oriented fish (geos, for example) won't 'care' as much as a fish clearly adapted for water depth (like angelfish with their tall fins). Vertically oriented fish (body plan or behavior) may have to adapt to seasonal low water in the wild, so it's not that they can't necessarily adapt to a shorter tank (within reason), but that doesn't mean they're most comfortable this way and the shorter tank can limit their behaviors, meaning you won't see some of the natural behaviors you will in a taller tank. This isn't just theory on my part, I've seen the difference with some fish in taller tanks, including how different species will separate into different levels of the tank, which can make a subtle (not everyone notices) difference in the feel of the tank and how it displays.
Imo you sometimes see inconsistency on this subject. Some, who emphasize tank width and would be critical of putting a 10 inch or longer fish in a tank only 13 or even 18 inches front to back, saying they won't be comfortable, or 'how will they turn around?' never say boo about putting a tall bodied or tall finned fish in a tank that's proportionally tight vertically.
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