Why does Nathan keep allowing Peter to hug him? (thus resulting in Peter stealing Nathan's flying power)

Nathan isn't aware of Peter's ability to replicate the power's of others. Peter tells Nathan that all he can do is fly in one episode. However, after seeing both he and Parkman use telekinetic abilities, it would be easy to assume that Peter has gained back his empathic mimicry (when in fact the audience knows otherwise) so Nathan would assume that Peter can already fly after having been in contact with him, rather than having to retake his ability each time.

So, with that in mind, a handshake or hug with his brother means little to him, and then Peter replicates his ability.

OK Peter asked Nathan i one episode asked him what the lastthing he saw himself do. And since Nathans reply was flying. Whenenver he comes into contact and needs to make a quick get away (which always seems to be needed) he simply gives Nathan a simple 'brotherly' hug or handshake and botta bing he fly's away

Because Nathan is fully focused on the task in hand, and is convinced he is doing the right thing. he really wants Peter to join him, because he still loves his brother, and is therefore willing to believe that peter wants to join him. They will always be family, so nathan hugs his brother in the hope Peter will change his mind.