In Peter Freyd's Abelian Categories, it is mentioned in passing that the category of abelian groups (more generally, $R$-modules for a ring $R$) satisfy the axiom AB5:
For each linearly ordered family $\{S_i\}_I$ in the lattice of subobjects of an object $S$, and $T$ is any subobject of $S$, then we have $$T \cap \bigcup S_i = \bigcup (T \cap S_i).$$
Isn't this completely obvious? Let $r \in RHS$. Then $r \in LHS$. Conversely if $r \in LHS$ then $r \in RHS$. (The only non-triviality is that the union of a family of submodules is a module in the first place, and this holds because the family is linearly ordered.)
I am confused because here https://ncatlab.org/nlab/show/Grothendieck+category#Kiersz they link to a 9 page paper that proves that the category of $R$-modules satisfies what they say is an equivalent condition to AB5 ('small filtered colimits are exact').
Yes, you are correct. The formulation of AB5 used on the nlab page you linked is substantially different from Freyd's (though it turns out to be equivalent) and so it takes a bit more work to prove it for modules over a ring. Note that the 9-page paper you refer to also starts by introducing all the concepts needed to formulate the statement from scratch. The actual proof of the final result (Proposition 4) is quite short and the important tools it uses from the previous 8 pages are just basic background on how to concretely identify all of the relevant category-theoretic notions in the case of modules over a ring (which is comparable to how you tacitly used the fact that the lattice of subobjects of a module can be identified with its usual lattice of submodules).