Only iCloud email contacts and gmail contacts support custom tags. If you are syncing contacts to any other email service you can’t use custom contacts. In particular Microsoft Exchange and Outlook do not, and neither do most other email servers.
The master data source for contacts (and Notes, Calendars, and Reminders) is your email server(s).
Go to Settings/Apps/Contacts and at the bottom it should show Default Account if you have more than one. This is the account that any new contacts will be saved on. If it isn’t iCloud or gmail then you can’t have custom labels.
If you don’t see it you have only one email account hosting your contacts. While in the Contacts settings tap on Accounts and it will list your accounts; see which one lists Contacts.