We know that when two groups are cyclic and have the same order then these two groups are isomorphic. If we know that one Group is cyclic and the other is not and they have the same order could we say that they are not isomorphic to each other?
Note that if $a$ is the generator of a cyclic group,i.e. $G=\langle a\rangle$, then $\phi(G)=\langle \phi(a)\rangle$, for any homomorphism $\phi$.
"If we know that one Group is cyclic, say $C_4$, and the other is not, say $C_2\times C_2$, and they have the same order, namely $4$, could we say that they are not isomorphic to each other?" Yes, we could. The property of being cyclic, or abelian (or being nilpotent, solvable etc.),is preserved under isomorphism. The idea of an "isomorphism" is, that all "algebraic" properties are preserved.