They would certainly not share an atmosphere.
If they were close enough to actually do that, they would invariably crash into each other, fusing into one.
Here is the actual scale of the distance between Earth and the Moon.
A twin planet system (one where both orbit each other) would likely be even farther apart.
They might have formed from the same local dust cloud, so their composition may be similar.
They might also both have life sharing the same DNA, due to panspermia. Life developing on one planet could be transported to the other via asteroid impacts.