I would drop the old semantics argument if it were me. Doesn't get you anywhere. Indeed, I always think of it as the 'tax disc' especially when I have to make my annual pilgrimage to the Post Office* to buy another one.
Bit like the "MOT certificate". That's what it's called, that's what it's always been called, you'd baffle your listeners if you called it anything else.
But "always"? No. When it was first introduced, in the 1960s, it was referred to as the "ten year test", soon shortened to "seven year test" and finally "three year". No mention of any Ministry. Only later did it become "MOT". And then, for a while in the 1970s, the MOT was absorbed into the good old Department of Environment, it became "DoE" for a while. Honest fact: I still have one or two ancient MOT certificates lying around that have "DoE" at the top instead of "MOT". But no-one actually called it that. "MOT" as far as I'm concerned.
Back to tax discs. Call them just that, "tax disc" and that conveniently obfuscates whatever the tax is actually for. And no-one can dispute that it certainly is a disc: i.e. a circle of paper with perforations round the edge...
*I insist on buying it over the counter. No mail-order or online stuff for me!