I doubt such a binary planet system would be stable.

Well, you asked for it. Any two binary orbiting bodies rotate around a common point, causing the entire system to rotate around it as an axis. This results in the planets alternating “closest to the sun” The sun’s gravity will naturally attract the closest one away from the farther one, and because the planets alternate, both planets will get larger and larger orbital radii. This will cause the two planet to eventually be ripped apart from each other.

This could be avoided if they were rogue planets, but the odds of two of those coming together are quite slim.