Mash made with butter (and garlic if you like, or sour cream, mustard, or cheese, etc.) will be rather different made from another starchy veg, and not just in colour. Common mashed veg might be sweet potato, or carrot and swede, and these, whether plain or flavoured, taste very different from mashed potato.
The same can be said for chips/fries/wedges: put the same seasoning on potato and sweet potato and you'll get different flavours. It's hard to separate the texture aspect completely for these, but the flavour difference dominates.
It's more a matter of serving tradition - yes potatoes are quite plain and enhanced by other things, but it's not essential to do so. The addition of butter to a baked potato or new potatoes is a lot to do with mouthfeel; the quantity is often too small to have much effect on the flavour. Also, even between potato varieties there's a difference in flavour, if you don't add so much seasoning that you mask it. This is particularly obvious with new potatoes.
In fact new potatoes are the dish in which it would be hardest to replace the potato with anything else (and it's not just because it's so simple - baked sweet potato, while not the same as baked potato, is rather similar and also has basically no other ingredients). If your friend reckons potatoes don't have their own flavour, steam some Charlottes or Jersey Royals and serve them those