Does the plaintext have to be at least as long as the key in Polyalphabetic substituion cipher In polyalphabetic substitution cipher, if we have a key $k$ (a string of letters) of length $n$ does the plaintext $m$ (also a string of letters) have to satisfy $|m| \geq n$? 

Also, side question, do we have to include a requirement that at least one letter in the key is different from the others? Because otherwise it becomes a simple shift cipher (Not poly at all)
I'm not aware of any classical polyalphabetic ciphers where the key could not be longer than the message.

For the Vigenère cipher, the key is, effectively, repeated to make it as long as the message.  There is no reason why it could not be repeated less than once, effectively discarding the unnecessary characters at the end of the key.

Similarly, in the autokey cipher, the key is dynamically extended by appending either the plaintext or the ciphertext message to it, ensuring that the extended key never runs out of characters.  If the message is no longer than the key, however, this extension is never needed, and the autokey cipher behaves exactly like the Vigenère cipher.

In fact, if the key for the Vigenère or autokey cipher is:


at least as long as the message,
composed of purely random characters, and
never used to encrypt more than one message,


then the cipher becomes an unconditionally secure one-time pad.  This is the most secure kind of cipher, being provably impossible to decrypt without knowledge of the key.  Alas, the strict requirements on the key &mdash; not the least being that it needs to be at least as long as the message being encrypted &mdash; make it rather impractical for most purposes.



As for your side question, obviously, if you deliberately choose only keys that consist of a single repeated character, then a Vigenère cipher becomes just a simple shift cipher.  However, if you're using a one-time pad with random keys, there is a (very small) probability that all the random characters will be the same, and that's perfectly fine.

In fact, if you threw out such keys, your key selection process would no longer be perfectly random, and so an attacker could potentially learn some information about the plaintext just by inspecting the ciphertext and knowing that the key cannot be one that has all its characters identical.