If you only have iCloud contacts (or only On My Mac contacts), you can create a Smart Group with the criteria, Card is not member of any group. Selecting the smart group will show the contacts that have not been assigned to a group.
There seems to be a bug that if you have both iCloud and On My Mac "accounts," it doesn't work correctly.
I couldn't find a way to add a contact to a category in iOS after creating it.
While simple, it's not well-known, and there has been a bug in it for quite some time. At least it works now, if you only have one type of contacts.
Using Smart Groups, Smart Folders, and Smart Mailboxes is pretty handy.
There is also a trick to see which groups a card is part of, with a contact selected, hold down the option key. All the groups that the card is a member of will highlight inthe Groups list.