The paper determines criteria of elementary equivalence for some classes of free groups with operators and free products with the length function. The case of a group with operators admitting rational coordinatization with a finite basis is completely analyzed. They are polycyclic, solvable groups of finite rank without torsion, and Chernikov groups. The concept of $\omega$-isomorphism of groups intermediate between elementary equivalence and isomorphism is important for the aspects of elementary equivalence of groups with operators and free product. it is proved that $\omega$-isomorphism of arbitrary groups of operators is followed by the elementary equivalence of the respective free operator groups (free products) with length function.