I've never seen tomato soup look like that, canned or not. It looks super diluted and very weird in colour. It looks more like a weirdly dark strawberry lemonade or something.
That has its own benefits, for sure. I have been known to use a press or even a toaster oven as well.
However, grilled cheese for me is similar to cooking an egg. It's the purest form of cooking and it's an art form that is easy to screw up.
The butter to bread ratio needs to be perfect and the bread needs a perfect toast in a pan. In rare cases I'll use some good cheeses, but American cheese is the only one that truly works. Butter the pan, not the bread. Keep the sammich moving at all times while it cooks. Timing is paramount for the toast.
If that makes the grilled cheese any better, I really don't know. There is, in measurable quantities, more love added to it, so by default, it's better.
It doesn't look like it was in contact with any cooking surface at all on the side we can see and we can't see the other side at all, so I don't know how you made the jump to panini, lol. It looks dry and sad.