The theory of your algebraic structure introduces a language for expressing calculations. For example, the theory of a ring introduces symbols $0,1,+,-,\cdot$, and we have a language using those symbols to express calculation.
Usually, we express things in this language unless there is a compelling reason not to do so.
Among the things ring homomorphisms must satisfy is requires $f(x \cdot y) = f(x) \cdot f(y)$. When you read that equation, you should read both instances of $\cdot$ as referring to the operation given in the theory of a ring; in this sense, both sides refer to the "same operation".
What's different about the two sides is the interpretation of the operation:
- In $f(x \cdot y)$, one interprets the operation in accordance with the domain of $f$
- In $f(x) \cdot f(y)$, one interprets the operation in accordance with the codomain of $f$
The (correct!) idea you are expressing is that these interpretations may be completely different functions between sets.
IF you were in a situation where you really did have cause to express arithmetic in terms of names for the specific functions that serve as the interpretations, then if the interpretation of $\cdot$ in the domain is a function we call $\circ$, and on the codomain is a function we call $\bullet$, then you would indeed translate $f(x \cdot y) = f(x) \cdot f(y)$ into the expression $f(x \circ y) = f(x) \bullet f(y)$.