A few weeks ago, the same phenomenon started to occur on one of my alts, whom rarely leaves home-base.
Each time I logged in, the characters name was missing. I discovered that the 'show name of own character' was switched off in Settings/Nameplates. I switched the setting on and that solved it but as soon as the character used a Wayshrine, the name disappeared once more.
This bugged me for several weeks but I eventually solved it. The problem in this case was down to an incorrect setting in the Tactical Combat addon, which was overriding the ESO "Settings" setting.
I am not in game at present, so I cannot give you an exact answer right now but I can approximate, which might help.
In TTC there is a selection of choices that can be made for your own group members.
One group deals with "Nameplates" and you can select whether they are shown or hidden. The name of my Char had somehow switched to the "Hide Own Characters" setting.
I switched the setting to the "Show" position and made sure that it was on the correct option in ESO's settings.
That did the trick and there has been no problem since.
I hope the above helps.