Is that a single plant or multiple seedlings? Whichever it is it/they need space of its/their own. I would hurry up and move the peppers because already the roots will be entangling and the pepper will be suffering from the competition.

That stuff next to the pepper plant, looks more like oregano than thyme, to me. I could be wrong !!

It's thyme, no doubt about it.

Why move the pepper at all. I wouldn't.

Looks like young german thyme plant(s) to me as well. I have thyme all over the place.

I don't think it will spread to fill the pot (at least not quickly). We have a german thyme that's at least five years old, and it got large and shrubby, but didn't creep like mother of thyme does.

It's not just a question of whether the thyme will fill the top of the container - it's what's going on underneath which is relevant. A pepper needs 1 - 2 gallons of space of its own, not a part share of a window box with what is effectively a small shrub. They will be competing for water and nutrients. "Relative large size of container combined with ample nutrition and water is needed for high yield production of pepper plant." See link.

I grow all my peppers in containers because of our cool summers. They get about 1.5 US gallons each.

Here is a link that might be useful: Pepper pot size

I would move the thyme to its own terra cotta pot with well draining soil. I say terra cotta because it will be a bit on the drier side (which thyme likes) in such a pot.

FataMorgana