Yeah it is absolutely possible, in terms of celestial mechanics.

In terms of the process of planet formation... it might be a bit improbable. But all that means is that you'd have to look for a while longer through the vast universe before you found it, right?

You could even paint a pretty clear picture by calculating the Earth-Earth Roche limit. Earth is not a rigid body, it is prone to deformation, but we could probably still use the rigid body equation if we give some extra padding. The Earth-Earth rigid body Roche limit is about 8000km so if you say it's ... I don't know... 2 or 3 times that you could probably avoid deforming your Earths too badly as they orbit each other.

In fact if you stick them at about 30 thousand km apart you could (I think...) preserve your 24 hour day, while also having a massive Earth in the sky of each Earth. But only from one part of each planet of course.

For people of this binary world, would be easier to establish a permanent presence in space than for us. But I think it would be harder to get to their Moon. (Assuming you keep our current Moon as it is, with maybe a slightly more wobbly orbit because of the proximity to the binary Earths around which it orbits.)

And of course exploring "the other Earth" would have been a major preoccupation throughout history. With all kinds of speculation and wild tales of what must lie above.

Really cool!