"Stage mother" in this case would mean "bossy, points out flaws, annoying, demanding." A stage mother is a woman who's child is doing a performance, but rather than let the child do it on their own the mother stays close and gives advice constantly. Normally this is considered overbearing for the child.

A "stage mother" is one who lives her dreams through the accomplishments of her child, generally associated with the stage (theater) or other prominent and visible acting professions. She is controls the child's career and is ever-present behind the scenes at performances. She is generally considered to be a nuisance to the professionals that the child must work with or report to, but is tolerated because she is in control of the child and makes decisions on the child's behalf. Any legal guardian can be a "stage mother", regardless of gender or age. It is because of stage mothers that some children opt to be legally emancipated.

I think so. Something motherly, but melodramatic.

"The stage" means "the theatre," or, by extension, the movies. A "stage mother" is the mother of a "child actor." A child actor's career must be managed by adults, usually the parents. Too often, it is an exploitative relationship, with the parents not acting in the child's best interest. The most extreme example is that of the Hollywood child actor, Jackie Coogan. He earned millions of dollars as a child actor and when he reached adulthood found that his parents had spent it all. As a result, in the 1930s, the "Coogan Law" was past, which protects child actors financially. However, there can still be emotional exploitation. The stereotypic "stage mother" is lives vicariously through the career of her child and micromanages everything.