Academically speaking, certainly the actual files are objects. They have attributes and you can perform actions on them. Doesn't mean FILE is a class, just saying, there are degrees of OO-ness to think about.
The trouble with trying to say that the stdio FILE interface qualifies as OO, however, is that the stdio FILE interface doesn't represent the 'objectness' of the file very well. You could use FILEs under plain old C in an OO way, but of course you forfeit the syntactic clarity afforded by Java or C++.
It should probably further be added that while you can't generate 'inheritance' from FILE, this further disqualifies it as OO, but you could argue that's more a fault of its environment (plain C) than the abstract idea of the file-as-object itself.
In fact .. you could probably make a case for FILE being something like a java interface. In the linux world, you can operate almost any kind of I/O device through the open/close/read/write/ioctl calls; the FILE functions are just covers on top of those; therefore in FILE you have something like an abstract class that defines the basic operations (open/read/etc) on an 'abstact i/o device', leaving it up to the various sorts of derived types to flesh those out with type-specific behavior.
Granted, it's very hard to see the OO in a pile of C code, and very easy to break the abstractions, which is why the actual OO languages are so much more popular these days.