Just a general note: Many islamic banks are open to anyone who is willing to accept this concept of banking and behave accordingly; and they do not ask about your religion. No particular information about FAB, though.
As @Forester said, Islamic banking is just banking by Islamic economic principles (quite interesting, worth reading up on). Such banks do not generally discriminate based on the applicant's religion.
Non-resident banking in UAE is very rare. You can pretty much forget about it unless you're very wealthy and/or you have a local company with significant local presence (office, employees, customers/suppliers).
And FAB just offers some islamic banking products, it's not like they follow islamic banking principles in their main banking operations. A friend of mine has a mortgage with FAB, and he pays interest on it: EIBOR + 3% or something. Very haram.
Yes, worth noting. It is in Dubai, I presume? (I guess that in Sharjah this really would not be welcome.)
no one cares really if westerner or not. Thats not a concept in banking, bc they must adhere to the allmighty correspondence banks (mainly usd ones).