Subscript is not special except of the way it is formatted. Let's replace it with e.g. h to clearly show what is the problem:

h[m, e] = 10;
m = 20;

m/h[m, e]

20 / h[20,e]

This was the first evaluation. You expected 10 instead of h[20, e] but since nothing prevents m or e from evaluation you end up with h[20,e]. And there is not any rule associated with h[20, e], only with h[m,e] (since it was created before m had value). You can confirm it with:

h[Unevaluated[m], e]

10

Now comes the second evaluation, which happens in different circumstances because m already has a value 20, which means that m will freely evaluate and:

h[m, e] = 10;

Will add h[20, e] = 10 to downvalues of h. Which will make

m/h[m, e]

evaluate in following steps:

m / h[m, e] ---> 20 / h[20, e] ---> 20 / 10 ---> 2

You can always use Trace to inspect evaluation steps or use ? to display rules associated with symbols. Where the latter was tricky in your case as it was about rules for Subscript you've created.