It seems that so long as a country is a secular democracy, religion should not enter into it.  I do think that if you can make the argument that Turkey shares a history with Europe, then you could make the same argument for Russia.  However, both have histories which also diverge from that of much of Europe.

I am wondering, in terms of contemporary geo-political concerns, what would obviate the inclusion of Turkey into Europe?  I should think that given it's proximity to the conflicts in the Middle East, there would be some concern about creating any "we've got your back" promises with Turkey.  In addition to any racism or Islamophobia, there may be practical concerns for aligning themselve too closely with Turkey.  Mind you, I'm not endorsing them, just suggesting that there might be less nefarious reasons...

Those who can make you believe absurdities can make you commit atrocities. -Voltaire