A digital signature is produced using a mathematical function that takes two numbers a and b from which the function produces a number c. The mathematical function is one which cannot be reversed, given c and a you cannot work out b.
So a can be the numeric value of a message being signed and b can be a number used as a private key that no one else can ever discover.
A separate mathematical function can take the message a, the signature c and another number d that is related to b and tell you if the signature is good. d is called a public key. You cannot mathematically work out the private key b from this public key d, so the private key b is always a secure secret that is not revealed by the signature or the checking process.
So the signature c of a message a is not usable as a signature for a different message, a different value of a produces a different value of c and the checking process will show this.
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