The 1970 FMVSS required the taillight to be the rearmost part of the bike; that taillight mounted directly on the alloy casting, the bottom edge of the fender stuck out a tiny bit beyond the light unit so BSA and Triumph came up with the tin extension to ensure the light position complied with that FMVSS. The extension was only fitted to/supplied with bikes made in the 1970 calendar year and sent to the US, as the FMVSS only applied to bikes made on or after 1st January; when BSA and Triumph switched to making 1971 model year bikes (September 1970?), they had a shorter rear fender and different taillight mounting.
Depending how "correct" your uncle likes his bikes, I just realised the tin extension is missing the part to mount the licence plate:-
View attachment 56648
... the strip kept the licence plate under the light.
The problem was the top edge of the licence plate had to be mounted on that strip and even if a new bike came with the strip for the bottom edge in the crate (apparently not all did

), because that second strip's two mounting bolts needed holes drilling in the fender, apparently often it was not fitted ...
However, if the strip for the bottom edge of the licence plate was not fitted, then the problem was, licence plate mounted only by its top edge, because these bikes vibrate a bit,

the vibrating licence plate broke the mounting strip off the extension ...

I think at least some states required payment for replacement licence plates; that was usually when the extension was removed and thrown away, the taillight mounted directly on the alloy casting, the new licence plate and bracket bolted to the fender ...
If your uncle wants one, those tin extensions are available new with the licence plate strip, I have the part number plus those for the bottom strip and its spacers off the fender, mounting bolt dimensions, etc., etc.
If you and your uncle know the following already, advance apologies. The TR5MX is a BSA B50, probably the "MX" (odd electrics, no lights, etc.) version; after BSA went bust in 1972, while BSA motorcycle production ceased, Triumph production continued, the "TR5MX" was an attempt to sell off existing B50's under the Triumph name. Aside, B50 parts bought by Alan Clews led to the founding of CCM (Clews Competition Motorcycles) -
History of the uber cool CCM Motorcycles.