Are disk identifiers important in Linux (CentOS)? [...]  Last,
googling suggests there is confusion between disk identifiers
and partition UUIDs. I am specifically asking about the
former.

That "disk identifiers" is not specific. Which one do you mean
among disk serial numbers, disk WWNs, disk label identifiers?
Also apart from partition identifiers there are also partition
lavbels, MD set identifiers, MD member identifiers, and filetree
labels and identifiers.  Not all of these are present in every
case.

It may be interesting to have a look at the contents of the
'/dev/disk/by-*/' directories.

As to the importance of any of these it depends on which
specific configuration goals you have. Each of them has some
advantages and disadvantages.