I have two ciphertexts (I do not know which encryption algorithm was used to create them) which both starts with the same 3-characters long word. This led me to believe that the same key was used for both of them and that I could exploit that to decipher it.
Are you sure that they are explicitly ciphertext? The reason that I ask is that quite often ciphertext are part of a file format or protocol. These protocols may contain other data than just the direct results of the cipher. They may e.g. be a Tag and Length in front of a ciphertext string as part of a TLV structure, or they may be a file magic, part of a non-random nonce etc. etc. etc.
Of course it could be that they are part of a ciphertext. In that case they may be generated using an identical key stream, and this key stream may be generated as part of a stream cipher. In that case you can try to solve it by looking up a many time pad as you currently do. However, it could also be that a plaintext aware stream cipher was used, and in that case you'll need to look at techniques to break the cipher.
If the ciphertext only contains printable characters (and not just hex or base 64) then that might hint at a classical cipher, which usually operates on one or a few characters at a time. Again, the analysis depends on the cipher though. Note that a stream cipher (with a randomized key stream) that uses XOR cannot just output printable characters, unless the result was explicitly converted to printable characters afterwards.
Finally, the chance that you get the same 3 bytes (not printable characters) is one in $2^{24}$ (about one in 16 million). It seems unlikely that this was generated by a (pseudo)-random function by chance, if this is indeed ciphertext. However, it is possible to deliberately cherry pick ciphertext if someone wanted to confuse you.
