They are now. The phone number was in my contacts, and the email was previously, but I have recently changed my email address for my apple ID so I just changed it in my contacts.
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Is there a "me" icon next to your name in the contacts list? My initials are next to my name in a gray circle if that's what you mean.
Follow
these instructions to set up "My Card" in contacts
I know that you may find this weird, but when I went to iCloud Drive on my Mac, contacts was in the list but not Messages. I checked "contacts." When I clicked on Options, neither one was listed there. I went ahead and turned them on in iPhone though.
Not weird, just an elderly version of MacOS. If you didn't find Messages, is iMessage in the list on your Mac? If so turn it ON.