There are geometric hypotheses that ensure the property you want. For example, suppose that $(M,g)$ has negative curvature. Then every $\alpha \in \pi_1(M)$ is freely homotopic to a unique geodesic representative $\alpha^*$. Usually people write $\ell_g(\alpha)$ for the length of $\alpha^*$. Finally, uniqueness of geodesic representatives implies that $\ell_g(\alpha^k) = k \cdot \ell_g(\alpha)$. This is just the beginning of an important area in Riemannian geometry. (When are geodesic representatives unique? What is the interaction between the metric and the variational properties of geodesics? And in a different direction: How does the fundamental group act on the universal cover? Etc.)

There are geometric hypotheses that ensure the property you want. For example, suppose that $(M,g)$ has negative curvature. Then every $\alpha \in \pi_1(M)$ is freely homotopic to a unique geodesic representative $\alpha^*$. Usually people write $\ell_g(\alpha)$ for the length of $\alpha^*$. Finally, uniqueness of geodesic representatives implies that $\ell_g(\alpha^k) = k \cdot \ell_g(\alpha)$. This is just the beginning of an important area in Riemannian geometry. (When are geodesic representatives unique? What is the interaction between the metric and the variational properties of geodesics? And in a different direction: How does the fundamental group act on the universal cover? What does the metric tell us about the algebraic topology of $M$ and the universal cover? Etc.)

It is amusing to contemplate all the ways in which the real projective plane, or more generally any closed manifold with finite fundamental group, differs from a negatively curved manifold.