Fix an Abelian category $\mathscr{A}$, and a homological spectral sequence $(E_{p,q}^r)$, so that $E^{r+1}\cong H(E^r)$, then, according to the ncatlab, we say that $E^r$ converges to $E^\infty$ if for all $(p,q)$, there exists some $r_0$ such that $r\ge r_0$ implies that $E^{r}_{p,q}\cong E^\infty_{p,q}$. Notably, however, this does not impose any condition on what the isomorphism is, or on naturality of the isomorphism.
I'm really struggling to understand why this is so. My intuition is that any concept in homological algebra that does not impose some sort of condition of naturality, or require an isomorphism to have some unique property, will end up being poorly defined. So why, for spectral sequences, do we allow there to be any isomorphism? If we wanted to define a functor from the category $$\mathbf{ConSpSeq}(\mathscr{A})\to\mathscr{A}$$ of convergent spectral sequences to $\mathscr{A}$ by sending $E$ to $E^\infty_{p,q}$ for fixed $(p,q)$, is it possible to define such a functor?
A better definition is that for $r \ge r_0$ all of the relevant differentials vanish, so that the homology of the page at that index is still $E_{p, q}^r$, which induces a (natural) isomorphism to $E_{p, q}^{r+1}$, etc. So there is a "stable value" of $E_{p, q}^r$ for large $r$, and you can define this to be $E_{p, q}^{\infty}$. This makes $E^{\infty}$ a functor on convergent spectral sequences.
The nLab definition just establishes the isomorphism type of the limit.