Outlook Contacts should be able to click Categories to sort.

[disccomp]

I know I can do a custom sort, but I should be able to sort just by clicking
the categories button at the top of the contacts list in Phone List View.

 

[Russ Valentine [MVP-Outlook]]

You never mentioned your Outlook version.
Most now use the "By Category" view for this purpose.

 

[Sue Mosher [MVP-Outlook]]

Right-click the Categories column heading and choose Group By This Field.
Because Categories is a keywords field, a sort would be meaningless.