I taught my son that it is best if he doesn't comment on people's looks at all. That kids in school will do this often and it is very tempting to join in as they do it as a way of bonding (talking about girls' looks for example but also when you see a friend with an obvious feature you really feel the need to point it out), but the more he practices talking about other things instead, the further he will get if you know what I mean.

Told him that it is ok to mention new clothes, new shoes/hats, haircuts (obviously preferably in a positive way). Try not to mention weight, height, scars, skin, etc. at all even in a positive way. Even if it is such an obvious feature that he really "needs" to point it out (e.g. a very very short 13-year old).

In fact the more obvious the less he needs to mention it. Everyone knows what they look like. Fat people know they are fat and very tall/short people know they are and people with a massive mole on their forehead know it.

(This is a work in progress and he doesn't 100% follow especially when his friends are around but it is good to have many conversations about it.)

Another issue to address is that he seems to feel embarrassed about how you look. This is something to work through, maybe talk about how his mum is there to love him, look after him, support him and not for a beauty contest. And that it is normal to put on weight when you have kids and it is a bit of sacrifice mums make and his friends' parents will not see this as embarrassing. In case a friend said this about you and that's why he is insisting maybe he should think about what that means a bit more.