I perceive a similarity in taste and odour between spearmint and peppermint, and I describe both as minty. The smell, taste and cooling sensation of peppermint are reproduced almost perfectly by pure menthol. But spearmint, despite being very closely related to peppermint, contains very little menthol, and it apparently gets its character from R-carvone, which is also found in caraway, which exhibits a vaguely minty aroma.
To me, lemon balm smells primarily like lemon, and not at all minty. Wild watermint (from which peppermint is obtained by hybridisation with spearmint) smells, to me, vaguely minty but also disagreeably vegetal, like lots of inedible greens do. Likewise catnip. And most other mint varieties sold in garden centres for culinary purposes (pineapple mint etc) smell, to me, similarly like weak mint with unpleasant off-notes.
On the other hand, pennyroyal and hyssop both smell unmistakably minty and pleasant to me.
What, then, do the two best-known mints have in common that makes them identifiably minty? Is it anything inherent to their makeup, or is it just mental association?
I've seen some Americans describe wintergreen as a type of mint. Botanically that's not true at all, and wintergreen oil's chemical composition is very different from the mints. I'm from the UK, where wintergreen is very rare as a food flavouring, and its smell is primarily associated with medicines and liniments. I don't perceive it as having a minty quality at all. I can recognise that it's pungent and herbal, so it has similarities to (for instance) eucalyptus, camphor, pine, rosemary, juniper etc, and in that sense, it's vaguely like the mints, but no more so than any of those other plants listed.
So I believe that when an American describes wintergreen as minty, it's because a) for them, the flavour is associated with the herbal candies referred to as "mints", and b) in broad terms, it's in the same sort of category. Is it the case that I similarly consider spearmint and peppermint both to be "minty", just because I'm used to using the word mint for both of them, even though they're very different chemically?
Addendum: here to illustrate the chemical differences are gas chromatography analyses.
Peppermint oil is primarily eucalyptol, menthol, menthone, menthofuran, menthyl acetate and iso-menthone
Spearmint oil is dominated by limonene, carvone, myrcene, beta-bourbonene; the spikes for menthol, menthyl acetate etc are tiny
(I believe that the distinctive characters of eucalyptus and wintergreen come primarily from eucalyptol and methyl salicilate respectively.)