My book is An Introduction to Manifolds by Loring W. Tu.
The last sentence here says
being an immersion or a submersion at p is equivalent to the maximality of rk $[\frac{\partial f^i}{\partial x_j}(p)]$
Is the above sentence then a generalization of the inverse function theorem, which is Theorem 6.26 and Remark 8.12 and which I asked about here, when we allow $\dim M \ne \dim N$? I think "full rank" is a generalization of "invertible" when our dimensions are not equal. Therefore, the sentence for equal dimensions gives "immersion or submersion" as "immersion at $p$, or equivalently submersion at $p$, or equivalently $F_{*,p}$ is an isomorphism" and "full rank" as "$F_{*,p}$, the map, or equivalently $F_{*,p}$, the Jacobian matrix, is invertible".
Edit: I edited the last paragraph and the title by asking about generalizations instead of analogies. I think the sentence is merely an analogy if we're given $\dim M \ne \dim N$ but a generalization if we allow both $\dim M \ne \dim N$ and $\dim M = \dim N$.