I would say this is impossible using inherent Revit features and awkward using Revit API. Tags wouldn't be my preferred solution for this in UI.
If I were doing it via Revit API I would use IUpdater to track wall location lines and adjusted associated text objects to match. Many complications would exist such as text element width to use and what to do when the wall length is very short (too short for text).
Overall I think this is one of those examples whereby if you adjust your approach to fit within existing Revit features you make your life easy compared with working against those features.
If the aim is to classify wall types then people generally do this by cut patterns/colours of materials making up wall layers. They then drag wall types onto a Legend view where a short section of wall is then depicted with those same colours to describe those wall types further. That for me is the most straightforward approach. As a drawing receiver that kind of approach also made my life easier when applying line loads to represent differing wall types. I see variations based on Legend (working out line load for 1m height of each type) and multiply by unsupported wall height based on where those types then are on plan. Similar could be said for understanding fire compartmentation etc.
