OED gives only adverb and conjunction in the relevant definition (effectively, In the way or manner described, indicated, or suggested).
In OP's context, "the thing told" is that Harry had never had any money in his life until a month ago. There's no need to see so as a pronoun referencing that "thing told". It's just an adverb, meaning...
in a manner embodying/conveying/signifying "the thing told"
("the thing told" will always have been already mentioned or implied by context).
Thus so doesn't refer directly to "all about having to wear Dudley's old clothes..." (but it does indirectly, since that clause simply expands on, and is a restatement of, "the thing told").
If there were other constructions where so appeared to function as a pronoun, it might be worth seeing it that way here. But I can't think of any such, so I suggest it's best seen as just another adverbial usage.
Note that to tell can be used in several ways...
But after a visit from my parents, I told. I was really feeling guilty because [blah blah].
(intransitive usage, no object)
No, that's not the way it was. I told the truth.
(transitive, direct object = the thing told)
He asked me what my mission was and I told him. (there are lots of told's in that link!)
(transitive, [in]direct object = the person told)
I told you the truth I heard from God
(ditransitive, with the two objects what was told, and to whom)
It's true we can often use "I told you so" and "I told you that" interchangeably, so possibly even some native speakers already associate so with "pronoun" functionality...
"He insulted my mother, so I hit him so!"*
(where the second "so" would be emphatically stressed, and probably accompanied by a gesture)
In that last example, you could perhaps say so = this [gesture], making it a "pronoun" usage. But I think it's more straightforward to call it adverbial so = like this, in this way.