iMessages
Why is my messages popping up as an iCloud? Instead of my phone number?
iPhone 12, iOS 14
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Go to Settings | Messages. The second line is "Send & Receive". Tap there. On the next page you should see that both your phone number and email address are listed as legitimate targets for messages sent to you. Below that is where you choose which of these, your email address or your phone number, should be used as your return address for threads you start anew.
Any threads that are ongoing and you simply add to, along with any threads originated from another party will forever have whatever ID was used for the initial message.