Is "so" a pronoun?

No. "So" is never a pronoun in English. In the case you cite it is an adverb, modifying "think".

In sentences like "I think so", so is an adverb.

This is not surprising if you think that the word yes is also an adverb.

I believe it is a demonstrative pronoun. See here and here. (Of course, the internet has much more to offer than that, as you should know.)

In these cases, the specific referent must be mentioned previously in the text for 'so' to work in such a way. 'This' and 'that', among other demonstrative pronouns, work in a similar way as well.

Feel free to correct me if I'm wrong since I've only been a student of linguistics for about a year and a half now.

The Cambridge Grammar of the English Language has three pages on anaphoric so. It concludes:

...its properties are unquestionably unique, and we do not believe that anything is gained by forcing it into one or more of our general part-of-speech categories.

Consider that so can stand for a noun (We are doctors. So are they.), an adjective (I’m sleepy. So is she.), a verb (Mine broke. So did hers.), a content clause (I thought that salamanders were amphibians. She thought so too.), a whole sentence or idea (The best things in life are free. I’ve always said so.) etc. Yet it can’t be the subject of a sentence. (In So am I, subject-verb agreement shows that I is subject, not so.)

We need to be careful with the idea of a demonstrative pronoun. Consider if we are standing in a store and I say, "I don't like that." You have no idea what I don't like and the sentence is meaningless unless I point to something. In this case, my communication includes a gesture; the gesture is the noun, and "that" is a demonstrative adjective modifying my gesture.

Generally what we call demonstrative pronouns are really adjectives that would modify a noun, but the noun has been dropped because the sentence is part of a chain of conversation and is understood. This is similar to a one word sentence, "Yes." The main part of the sentence has been dropped. This does not mean that "Yes" is both a subject and verb simultaneously forming a grammatically complete sentence, nor that it is a demonstrative pronoun. It means that in conversation complete sentences are not needed in order to be understood.

Demonstrative pronouns, if they actually exist, should not be used in legal contracts or legislation because it results in ambiguity. In conversation they are fine, but then, a lot of things are.

1 It looks like rain. -- No, I don't think so (I don't think that it is so/in such a way).

2 We are doctors. -- So are we. (We are doctors as well. Compare German ebenso: One might say in German: Ebenso sind wir Ärzte. It is not idiomatic today, but everybody would understand it. Today one would say: Wir auch.)

The use of "so" in "So are we" for "We too/We as well" is a bit curious. But it is a common manner of answer. It is a bit cumbersome to reconstruct a longer formula with "so". One would have to make two sentences: So/in the same way it is true for us. We are doctors too.

Please take all this as a hypothesis of mine. An attempt to make such uses of "so" better understandable and to show that we actually have the normal use of "so" meaning "in such a way".

It's a pronoun used to substitute for a clause or sentence, which may be understood. For example:

You'll stop because I said so.

If we look at 'so' the way we see the pronouns like something , nothing and everything that succeed the verb 'say ' , we might tend to suppose 'so' to be a pronoun like them but 'so' is to be thought as conveying the manner of the verb , say in the sense of ' the same way ' or ' in such a way ' but not as a replacement to an object which is usually a noun or pronoun as shown in the sentences below Rabbi said something . Samiya said an apology to me . In the sentences , Rabbi said so . Samiya thought so 'so ' is just telling us how Rabbi said or how samiya thought Or giving us a sense that they chose the path similar to what was mentioned earlier but it may not be what they exactly said or thought . If it should be so , we do have an alternative phrase to be used as a pronoun , which is ...' the same ' I think the following sentences will be of some use to you in clearing up this ambiguity Samiya thought so (samiya thought in such a way as was mentioned earlier ) Samiya thought the same (samiya thought exactly what was told earlier.