NOTE:Be aware I am using the latest MacOS/iOS/iPadOS beta on all of my devices, and the data on all of my devices is tightly synchronized which could bias my results.
That said, I have tried every permutation I can think of and cannot duplicate your results. In fact, no matter what address I use (phone number or email) or which device I send the text from they all end up in the same Conversation with my name and picture at the top as the sender.
However, I did have an old unused address offered when I was sending some of the texts, because part of my current address was duplicated in the unused address. When you send a text message your identification is typically drawn from your entry in Contacts/AddressBook. You might search Contacts on each of your devices for your name, your email address, and your phone number and see if there are multiple entries, and if so consolidate those into a single entry. (the one I found contained ONLY the email address and it contained only a part of the address I was entering.) Something like that could account for your situation.
