What exactly makes a carbon atom "α" in a protein residue?
This nomenclature is due to the fact that amino acids are carboxylic acids.

Near the carboxylic acid moiety, the carbon chain is unbranched and simple, so the positions are named like an unbranched, simple aliphatic carboxylic acid.

The carboxylic acid ($\ce{-CO2H}$) is not indicated with a position. But the carbon immediately next to it is $\alpha$. The one next to that is $\beta$. That should be enough to explain the $\alpha$ in your context because that carbon is the one immediately next to the carboxylic acid. In a protein or polypeptide, the carboxylic acid is most frequently converted to an amide in a peptide bond, but the Greek letter nomenclature is unchanged.

This nomenclature is used in other cases for example, with an $\alpha$,$\beta$-unsaturated carbonyl.

The most publicly visible use of this nomenclature is when we use $\omega$ to denote the end of the carboxylic acid chain. This is the origin of the term $\omega$-3 [unsaturated] fatty acid.