If I sign the same message several times, the bitcoin-qt returns the same signature.
It does not use a random seed for signing, such as signing transactions?
No, message signing is different from transaction signing.
Message signing has three parts: (1) Address (with privatekey), (2) Message, and (3) Signature.
When signing a message, you are providing proof that the message comes from that bitcoin address
(and thus you have control over that privatekey).
If you change the address you are signing from, your signature will change.
There is no random seed in basic message signing, since that would defeat others ability to verify it.
The same signing algorithm is used because really it is all just signing bytes of data.
If I sign two transactions with the same seed, I expose my private key.
The same thing happens with the message signature?
Yes, that can happen. But that will not happen here because the signatures are identical. You only expose the private key if the R values are the same but the S values are different. In this case, because the signatures are identical, then both R and S are identical so it doesn't matter.