I know this is wrong, but I don't know where I'm wrong (though I'm sure it's something stupid):
An isogeny between two elliptic curves defined over a finite field is a surjective homomorphism between their groups. Tate's Isogeny Theorem in this case tells us that the numbers of points on the two groups are equal (and, thus, the two groups have the same order). But, being finite, surjectivity implies injectivity. Thus, the homomorphism should always be an isomorphism.
Is my error that the homomorphism is surjective only for the curves defined over the closure of the fields (which aren't finite)?
An isogeny is surjective over the algebraic closure of $\mathbb{F}_p$. It is not necessarily surjective over $\mathbb{F}_p$ itself.
To give a simple example illustrating the issue: the map $f(x) = x^2$ is surjective over the algebraic closure of $\mathbb{F}_p$, but it is not surjective over $\mathbb{F}_p$ itself. In fact this example is more or less exactly what is going on in the isogeny setting.