Q:

When using RSA algorithm, is it a risk if plaintext and its ciphertext are identical? if yes why?

Is it a risk if RSA produce exactly same ciphertext for the given plaintext?

A:

RSA encryption with a 1024-bit public key encrypts messages up to 117 bytes, and the corresponding ciphertext has length 128 bytes. More generically, for all key sizes and with standard PKCS#1 padding modes, the ciphertext is always longer than the plaintext. Thus, they cannot be equal to each other.
If you consider post-padding input (i.e. the 128-byte string which goes under exponentiation) then a fixed point (a number which goes unchanged through the exponentiation) may expose parts of the private key; however, you will not hit it by chance. Or, rather, you will hit it only with negligible probability (you can force it if you wish, but nobody can prevent you from revealing your own private key anyway). An attacker gains no extra information by such a random occurrence, because such an equality can already be tested by the attacker... so the answer is: no, there is no risk in that.

