Is me a concrete noun

No, the Word 'me' is not a noun. The Word 'me' is a pronoun, a Word that takes the place of a noun in a sentence.

The pronoun 'me' is a personal pronoun, a Word that takes the place of a noun for a specific person or thing.

The pronoun 'me' is a singular, first person, objective pronoun, a Word that takes the place of a noun (name) for the person speaking as the object of a verb or a preposition.

Example: My niece gave me a big smile.

The corresponding singular, first person, subjective personal pronoun is 'I'.

Example: I smiled right back at her.