Corollary 4.11 of Silverman's book The Arithmetic of elliptic curves (p. 73) says
Let $$ \phi:E_1 \rightarrow E_2 \text{ and } \psi:E_1 \rightarrow E_3 $$ be nonconstant isogenies, and assume that $\phi$ is separable, If $$ \ker \phi \subset \ker \psi $$ then there is a unique isogeny $$ \lambda: E_2 \rightarrow E_3 $$ satisfying $\psi = \lambda\circ \phi$.
Does this result extend to abelian varieties of higher dimensions?
Yes it does. If $\phi : A_{1} \to A_{2}$ and $\psi: A_{1} \to A_{3}$ are isogenies of abelian varieties as in your question, then you may form a quotient $A_{2}/\phi(\ker \psi)$ with quotient map $\pi : A_{2} \to A_{2}/\phi(\ker \psi)$. The composite map $\pi \circ \phi$ then has kernel exactly $\ker \psi$, and so its image is $A_{2}/\phi(\ker \psi)$ is isomorphic to $A_{3}$ by Corollary 1, pg. 118 of Mumford's book on Abelian Varieties. Post-composing $\pi$ with this isomorphism then gets you an isogeny $\pi' : A_{2} \to A_{3}$ such that $\psi = \pi' \circ \phi$.