Sweet pea pods are narrower compared with their length than mange-tout pods, and they are slightly hairy. And they are a bluish green rather than a 'pea' green. There's nothing in that photo that would make me doubt it was a mange-tout, (and it is certainly not an ordinary sweet pea) but I'd never say to anyone 'go ahead and eat it' without seeing the plant itself.

I'm not certain without going outside and looking, but don't sweet peas have winged stems?

It's the shape of the flower that tells you the plant is in the pea family, so similarity in the flowers is the least thing you should worry about. All the pea family have the same flowers - peas, beans, laburnum, wisteria, right down to clovers.

So the thing to do is to compare the stems and leaves and pods as you have.
Mange-tout peas look very much like ordinary culinary peas, so you could reassure yourself by taking a spray of your mange-tout and comparing it with some peas on an allotment or in a neighbour's garden.