How common is this? Does it suggest a particular style or register?

It is a very common use of “ou” in a list. No particular register, just to express the fact that you can choose between the elements of the list.

Is it only used in lists that could mean something like “such as”? If not, where else?

Not like “such as”, more like “One way or another”, not used to compare but to give another choice.

Would “et” sound strange here? For example, would “et” suggest that the list is exhaustive?

If you used “et” here, you would suggest that every element of the list is part of the same group.
Exemples : Clever, cool and big are adjectives. You can pick clever, cool or big to describe something or someone.
“Et” would not sound strange, it would change the meaning of the sentence.