Clarifying the homomorphism's definition?

And your intuition is correct. Usually, the homomorphism is between two algebraic structures with distinct operations.

Does it mean that homomorphism suppose to work only with algebraic objects sharing the same operation?

No. It means that, when working with algebraic objects, we usually share the same symbol for different operations.

Herstein's book (p. 67) explains this issue in the context of groups:

Definition. Let $G, G'$ be two groups; then the mapping $\varphi:G\to G'$ is a homomorphism if $\varphi(ab)=\varphi(a)\varphi(b)$ for all $a, b\in G$.

In this definition the product on the left side - in $\varphi (ab )$ - is that of $G$, while the product $\varphi(a)\varphi(b)$ is that of $G'$.

Analogously, for a general homomorphism $f(x \circ y) = (f x) \circ (f y)$, the symbol $\circ$ on the left stands for the operation in the domain, while the symbol $\circ$ on the right stands for the operation in the codomain.

Yes, you are right. Take, for instance, $A=B=\mathbb{R}$, and take the usual sum ($+$) on $A$ and the usual product ($\cdot$) on $B$. Then a homomorphism between the two structures $(A,+)$ and $(B,\cdot)$ is a function $f:A\longrightarrow B$ such that for each $x,y\in A$ we have \begin{equation} f(x+y)=f(x)\cdot f(y) \end{equation} An example of such function is the exponential, $f(x)=e^x$.