After having read GWB's book
Decision Points and his very revealing stance vis a vis Tony Blair and Vlad Putin, I'd have to say that Bush looked at his allies -- Blair -- from the standpoint of a rival. Allies? Yes. Trusted comrades? Absolutely not. Why not? Because Blair was liberal in thought word and deed and GWB....well, let's just say that GWB thinks of himself as a conservative, but that's just balderdash (I like that word -- it sounds kinkier than "horseshit"

).
At least at first.
Afterwards, that relationship warmed considerably to the point of real, honest-to-God friendship.
Putin was a whole 'nother critter.
GWB made the mistake of actually believing Putin and what he said. And I don't ever once think that GWB looked at Putin like an ally -- though I think he was thinking it might be possible. Though I do think that GWB was thinking it might be possible to be personal friends in the same way that he and Blair were friends.
Putin wound up sticking the shiv into GWB's ribs, so that angle died fairly quickly. He should've known -- GWB had momentarily forgotten that the Great Russian Bear does not, and cannot, ever trust anybody or anything that comes out of the west. It's in their blood.