Does anyone know why 'ich' is being used here, and not 'mich'? Would the sentence 'Dieses Mädchen bin mich' be wrong?
In sentences using verbs such as sein, bleiben, etc., we have predicate nominatives. This means that the verb "links" the subject with an attribute of said subject ("renames the subject"), and in German then the two linked entities agree in case, i.e. in nominative case.
If you inverse the word order, this becomes clearer:
Ich bin dieses Mädchen.
ich is the subject, dieses Mädchen is the noun phrase, both are in nominative and linked by the predicate (verb) sein.
"ich" is correct, and "mich" is certainly incorrect. If you look here, you can see that "me" in English is technically, or rather traditionally incorrect, though obviously everyone would say "me", it's common usage.
In German, "mich" (one equivalent of "me") would be incorrect, and is not common usage. "mich" is accusative, the verb "sein" (to be) never takes the accusative case - only the nominative case, i.e. "ich".
I suppose since English used to say "It is I" instead of "it is me", Germans just kept using the old form? Similar to how "Thou" used to be our informal version of "you" but because of capitalism and not knowing people's class statuses definitively, as well as thou becoming a popular way to insult your betters, we just changed our vocabulary.
Yes, German kept the old forms.
It also didn't split up "the" and "that", for example, or "to" and "too".
I think the correct translation of "Dieses Madschen bin ich" should be: "I am this girl" .
If we would like to say "this girl is me" it should be "Dieses Madschen ist mich." Bin ich richtig, oder?
You are not correct.
"to be" does not use the accusative case, so ist mich would be wrong anyway. And Madschen is not a German word.
It connects two nouns which refer to the same thing -- and if one of them is a pronoun, it decides the form of the verb. So only Dieses Mädchen bin ich is possible.
so would 'Dieses Mädchen ist ich.' be wrong and, if so, why?
Yes, it would, because the verb has to agree with the personal pronoun ich rather than the noun dieses Mädchen.
English and German work differently here. Another example: English says "It is I" or "It's me" but German has ich bin's rather than es ist ich or ich ist es.