What aspects of a fridge most affect the quality/taste of food?
There are a number of other threads here with some bearing on this question, including:

What is the ideal fridge temperature

Can food spoil at cold temperatures?

How important is isolating food in a fridge?

Does having spoiling food in your fridge cause other food to spoil faster?

What we can glean from all of these is that temperature control is critical, and can vary between different models.  You want a refrigerator that can reliably maintain a consistent temperature without wide temperature spikes, and you probably want a relatively low differential between different parts of the box (so that, for instance, items in the bottom compartment don't freeze while items on the top shelf spoil).  Different compartments can help in isolating foods from one another, but they would be especially valuable if their temperature and humidity could can be independently adjusted to allow for minute control.

Also, keeping your fridge clean is important, so that mold, bacteria, and so on can't collect and spoil your fresh new produce; thus you want a model that can be readily disassembled and scrubbed. Easily removable shelves, compartments, and so on would be beneficial, and they should be made from easy-to-clean plastics or glass with minimal texture.  At the same time, these need to be sturdy and adjustable, so that you can thaw that 20-pound turkey without having to sacrifice too much other space or breaking a shelf.

Speaking as not just a chef, but as a home bartender, I also want adjustable space inside the door(s) so that I can store a variety of differently-sized bottles upright.  But that's more for convenience and efficient storage than it is about taste or quality.