If the key and IV are identical then the first intermediate value after the block encrypt that is XOR'ed with the first byte is also identical between two encryption operations. That means that if the first byte of plaintext is identical then this will result in identical ciphertext, leaking information to a possible attacker.
Of course, this ciphertext is also propagated into the shift register that currently holds the IV. One byte of the IV is shifted out to the left (MSB), and ciphertext byte is placed at the right (LSB). So now the next encryption will again result in identical intermediate value. This means that the next identical plaintext byte will also directly leak data, and so on and so forth. Of course, if you have many messages then you can make many pairs, so the leakage of data is more likely.
Only if the plaintext bytes differ between messages then the propagation stops. However, these final bytes form a problem in themselves. The differing plaintext have been XOR'ed with identical intermediate values to create the ciphertext bytes. That means that the XOR of the ciphertext bytes results in the XOR of the plaintext bytes. This may well directly leak data, and again, the more ciphertext bytes are known, the more combinations can be made.
On the bright side: the IV stored in the shift register. As it is only used as input to the block cipher it is likely that the value of the IV and thus key is relatively well protected.
If low level side channel attacks are possible then it may be possible to identify some key bits during the shift operation, but you'd probably need a lot of operations before any IV / key data could be extracted. As I've been surprised by side channel attacks before I think it should be considered though.
Not only shouldn't you use key data as IV (using a hash over the IV would already have been better), but you should also not re-use the key for different purposes.
I would be very skeptical to the security of the protocol you've described due to these cryptographic bad practices - although the use of CFB-8 mode is probably already enough indication that the designers of the protocol didn't know what they were doing.