I'm 100% with the OP, and also with the poster who said that the BIG label should have read 'Mozarella with flavoured mayonnaise'. Because that's what the list of ingredients posted in the label up-thread shows us. (I presume that refers to the wrap.)
By law, ingredients have to be listed proportionally, with the largest percentage of ingredients first. So far as I can see, the M&S label puts rapeseed oil (used in most 'industrial' mayos) and palm oil (nasty taste IMHO and often bad for the environment) and also eggs (used in mayo but not pesto) ahead of olive oil (used in 'proper' pesto) in the list of ingredients. There is also 'basil-infused sunflower oil' in the list - am not sure what that is doing. Also, you will see that rocket - not mentioned on the wrap front labels at all - comes AHEAD of basil in the list of ingredients. So the ground-up green leaves mingled with the mayonnaise will be more rocket that basil, apparently. And there are additives such as glycerol, not found in real pesto or real mayonnaise.
The whole label is a lesson in why ultra-processed foods are bad - for our understanding and appreciation of real food, for the planet - and for our health.
As yet other posters have said, 'real' food has few ingredients, ideally of good quality. Could not agree more.
I'm not getting in to the debate about what 'real' pesto is. All I would like to say is that you can make a fresh and zingy 'real' basil-leaf dip - takes less than 5 mins - this way:
A bunch basil, washed (I use Lidl's - last time was 55p, a bargain).
Good slosh olive oil.
Juice of one small lemon, or half a big one.
A very small clove garlic
A heaped tablespoon, approx - adjust to taste - ground almonds or 'raw' cashew nuts. I prefer almonds.
Salt and pepper. Ideally, both freshly ground.
Put all in a bowl and blend. I use a stick blender.
The smell of the fresh basil is heavenly. If you eat cheese, you can grate it on top of whatever (pasta, broccoli, green lentils, sliced tomatoes etc etc ) your basil-leaf dip is mixed with. I do eat cheese, but I prefer it without.
Any leftovers will keep, covered, in the fridge for a day or two.