A couple of points on terminology. In your question you state that they will contain the same files. So their content is identical.. This statement is ambiguous at best. When discussing high-level cryptography, files and content have little meaning. Instead, there is only plaintext and cyphertext.
Lets reword your question for clarity:
- Given two or more encrypted volumes initialized with the same passphrase, and each having the same size
- With an identical set of files placed into each after initialization
- Does an attacker gain any advantage by independently knowing that each volume, when presented with the correct decryption key, will produce an identical set of files?
If the above is correct, then the short answer is either "no", or "a very slight advantage". A longer answer depends on the exact encryption scheme used.
For the short version, let's use VeryCrypt for illustration. When a VC volume is initialized, you are asked to move the mouse cursor randomly to generate entropy. This entropy will be very different for each of your volumes and is independent of using a repeated passphrase. The entroypy serves two purposes. First, for many encryption schemes it is used to generate the IV (initialization vector). The IV is encrypted or combined with the passphrase and is used in a sequence of transformations on the plaintext. Part of the output of the previous sequence or block is used as the input for the next. This means that even though the plaintext may be identical, given different IVs, the cyphertext blocks will be very different from each other.
Now, if you were merely encrypting a single file or set of files as opposed to placing them into a volume, you could expose your data to various types of plaintext attacks including the one described by @arthurmilton. However, when using a volume or container, the entropy and IV are also used to fill the container with random data. This random data is encrypted using the same scheme and the random cyphertext that is produced is practically indistinguishable from the real cypher text belonging to the files. I say practically indistinguishable because depending on the scheme being used, there are theoretical ways to increase the probability of identifying the random cyphertext from the file cyphertext, but AFIK there has never been a real-world attack of this nature on a well-tested encryption scheme.
Taken together, these two things mean that an attacker gains nothing or almost nothing from her knowledge, since any comparison between the containers would require too vast a number of calculations to extract any pattern.
There are a few theoretical weak points (and assumptions) that interfere with above. First, we are relying very heavily on the idea the randomness generator will produce random enough sequences that an attacker who is blind to the entropy source(s) (the random mouse movements and other factors), won't be able to detect any patterns in a computationally feasible manner. This source of randomness is one of the harder problems to fully solve in encryption, but it appears that the generator in VeraCrypt is "good enough" for now. Second, we are assuming that the container is larger than the size of the files being placed into it. When combined with poor entropy generation, less random cyphertext, could make it slightly easier to identify the cyphertext that belongs to the real files. (I'm somewhat misusing the distinction between random and non-random cyphertext, but it helps illustrate the point). There are other counters to this issue, and VeryCrypt does not require or even recommend that some ratio of files to available space be maintained in their containers.
Second, we are assuming that the encryption scheme uses a transformation mode with known protections against reuse attacks. This is a very dense area that I don't have the expertise to simplify.