In practice, yes, an identical cryptographic hash means the files are the same, as long as the files were not crafted by an attacker or other malicious entity. The odds of random collisions with any well-designed cryptographic hash function is so small as to be negligible in practice and in the absence of an active attacker.
In general, however, no, we cannot say that two arbitrary files having the same hash definitely means that they are identical.
The way a cryptographic hash function works is to take an arbitrary-length input, and output a fixed-length value computed from the input. Some hash functions have multiple output lengths to choose from, but the output is still to some degree a fixed-length value. This value will be up to a few dozen bytes long; the hash algorithms with the longest output value in common use today have a 512-bit output, and a 512-bit output is 64 bytes.
If an input to a hash function is longer than the output of the hash function, some fidelity must be removed to make the input fit in the output. Consequently, there must exist multiple inputs of lengths greater than the length of the output, which generate the same output.
Let's take the current workhorse, SHA-256, as an example. It outputs a hash of 256 bits, or 32 bytes. If you have two files which are each exactly 32 bytes long, but different, these should (assuming no flaw in the algorithm) hash to different values, no matter the content of the files; in mathematical terms, the hash is a function mapping a 2256 input space onto a 2256 output space, which should be possible to do without collisions. However, if you have two files that are each 33 bytes long, there must exist some combination of inputs that give the same 32-byte output hash value for both files, because we're now mapping a 2264 input space onto a 2256 output space; here, we can readily see that there should, on average, exist 28 inputs for every single output. Take this further, and with 64-byte files there should exist 2256 inputs for every single output!
Cryptographic hash functions are designed such that it's computationally difficult to compose an input that gives a particular output, or compose two inputs that give the same output. This is known as preimage attack resistance or collision attack resistance. It's not impossible to find these collisions; it's just intended to be really, really, really, really hard. (A bit of a special case of a collision attack is a birthday attack.)
Some algorithms are better than others at resisting attackers. MD5 is generally considered completely broken these days, but last I looked, it still sported pretty good first preimage resistance. SHA-1 is likewise effectively broken; preimage attacks have been demonstrated, but require specific conditions, though there's no reason to believe that will be the case indefinitely; as the saying goes, attacks always get better, they never get worse. SHA-256/384/512 are currently still believed safe for most purposes. However, if you're just interested in seeing if two non-maliciously-crafted, valid files are the same, then any of these should be sufficient, because the input space is sufficiently constrained already that you'd be mostly interested in random collisions. If you have any reason to believe that the files were crafted maliciously, then you need to at the very least use a cryptographic hash function that is currently believed safe, which puts the lower bar at SHA-256.
First preimage is to find an input that yields a specific output hash value; second preimage is to find one input that gives the same output as another, specified input; collision is to find two inputs that yield the same output, without regard to what that is and sometimes without regard to what the inputs are.
All that said, it's important to keep in mind that the files may have very different data representations and still display exactly the same. So they can appear to be the same even though their cryptographic hashes don't match, but if the hashes match then they are extremely likely to appear the same.