This also happens to me. I am a streamer, but I do not have streamer mode, or Replace gamertags/names/etc active. I have flipped them back and forth multiple times. My experience is as follows:
Start game. My ship name (HMS Shadowlands) is present. When approaching ship, flash message saying "HMS Shadowlands, Captained by NikoyakaSanzoku" appears, confirming that the name is accurate. Set sail with a crew. After an indeterminate period of time, I happen to catch my Name Plate in my field of view and the name has changed. The crew can still see my original nameplate. The next time I set sail the name has returned to HMS Shadowlands... Then repeat as above with the name change occurring during the game for no reason I can understand.
I have checked settings. I have uninstalled, and reinstalled the game. I have toggled almost every setting on and off again. Nothing corrects it. The name HMS Shadowlands was chosen because it means something to me, and I am the only one that cannot see it.
For reference, none of the Usernames I can see (my crew, for example) are changed. They show up accurately, meaning that as people names and ship names seemed to be linked in the "Show gamertag" group of settings, it definitely is not that.
I am frustrated beyond belief at this, given how long we waited to have this customisation option.
I work in Software Development (UAT Test Lead for a Multinational Educational Organisation) so I understand the intricacies of the process. However this has been going on for months, with a LOT of people reporting similar problems. If the Devs cannot replicate the issue, perhaps it is time to reach out to the community directly?
(Edit: I am playing on a Windows 10 Desktop PC, with more than enough RAM (32GB) and Graphics power (12GB) to run SoT at the highest settings. So it is unlikely to be a system issue. Happy to be corrected though.)