adjective comparison

Mary is fatter than me / her / him / them / us / you ~ everyday informal English

Mary sings better than me / her / him / them / us / you. ~ everyday informal English

Mary is fatter than I am / she is / he is/ they are/ we are/ you are. ~ correct

Mary sings better than I do/ she does/ he does/ they do/ we do / you do. ~ correct

A pronoun directly after “than” has the object form.
I am taller than him.

But if there is a verb after the pronoun, then the pronoun has a subject form.
I am taller than he is.

You may hear a sentence like "I am taller than he " with a subject pronoun at the end. This is formal and old-fashioned. It is more usual to say I am taller than him or I am taller than he is.

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John likes chocolates more than me.

You could convey the wrong meaning if you weren’t careful.

John likes chocolates more than toffee.
This obviously means John likes chocolates more than he likes toffee.

John likes chocolates more than me. (~ ambiguity)
Like is a transitive verb in this sentence and it takes an object.
Since “me” is the object pronoun, similar to “toffee”, the sentence seems to mean:
John likes chocolates more than he likes me.

It is better to express more clearly without ambiguity.
John likes chocolates more than I do. ( ~ clear; " I" is a subject pronoun).
John likes chocolates more than he likes me. ( ~ clear)

Your sentence should state what things are being compared to avoid ambiguity.

Reference: Practical English Usage

Before I write my answer, I'd like you to think of the sentences below:

John likes chocolates more than me.

Does John eat more chocolates than I do?
or
Does John like chocolates more than he likes me?

Object pronoun...

because "Mary" is the subject and "I/he/she/they/you" is the object in the structure of the sentence in relation to the subject.

Unless the writing of the sentence is that Mary and "I" are equal "subject", only then subject pronoun will be used.