Does "I believe as strongly as anyone that" mean "as strongly as the most believing person"?

I've always been somewhat confused by this phrase because the tone of the context, which is often refuting someone else, is that they believe in it fully--to the extreme--but the actual words suggest that they mean they are in agreement with the general broad belief of society (which do not usually overlap in these usages).

The short answer to the question, as formulated in the title, is yes. If one doesn't find that obvious, here is a way of thinking about it that might help one to see it.

The phrase as strongly as is sometimes used to mean that the strength of the first item is exactly equal to the strength of the second item. Sometimes, it is, however, used to mean the strength of the first item is at least as great as the strength of the second item, i.e. that the strength of the second item is not greater than that of the first. It is in that latter sense that the term is used here. (The existence of such two meanings is not peculiar to this term; the same ambiguity exists in as large as, as fast as, as smart as, as rich as . . . .)

Now the sentence says that I believe this not just (at least) as strongly as this or that specific person, but as strongly as anyone. The sentence, so to speak, invites you to go ahead and pick any person you please, and tells you that, whichever person you pick, it will turn out that I believe it at least as strongly as that person. As you can pick any person, it is possible that you will pick the person whose belief is the strongest (among the relevant beliefs of all people other than me). If the sentence is to be true, it then has to be the case that I believe it at least as strongly as that person. The sentence is thus logically equivalent to 'Nobody believes more strongly than I do that . . .'.

Such declarations are often followed by 'but . . .', that is they often introduce some qualification or limitation of the belief, but that is not a part of their literal meaning.

To borrow from Jason, hat tip, Anyone includes everyone and makes little distinction. Doubtful or not they are included, no contradiction is intended by including everyone. It is a way to say "I too, believe. Don't hassle me." What must be understood is that this phrase is usually used to preface a specific condition the speaker wishes to stipulate.

"Despite my agreement and dutiful belief along with most people I think there is another condition requiring clarification that a non-believer might raise." There can be confusion when the speaker who has just agreed with the belief then steps into the critical non-believer role to get some questions answered. The phrase is used to introduce detailed doubts or exceptions to the idea.