You've reminded me of the Life Well Lived - spotify podcast I listened to today. They interviewed Claude Steele a US social psychologist. Steele spoke about stereotype threat, the idea that underperformance in an activity can be brought on by the threat of confirming a stereotype about the participants own context, such as social standing, race, gender, etc).
Steele cited a mini golf study where the black and white 'elite athletes' were brought in to play one at a time. Before each started they put the white athletes under stereotype threat by saying "This mini golf course is really a standardised test of natural athletic ability. Do the best you can." By making this statement it was posited those subjects had an added mental load. To describe differently were multi-tasking. They were trying to play the course, but also thinking about whether they're confirming those stereotypes about white athletes being worse in natural ability than black athletes. This in turn makes those players play worse than average. Anyway sounded an interesting idea.
The way it relates to what you're saying is maybe Albanese being fixated on his class background is because he's feeling that stereotype threat in so many situations he dives into as PM.
Maybe the defensiveness is a result of his fear of confirming the stereotype that only 'elite' schooled or wealthy children can grow up to lead a nation. This distracting thought in turn means his mind has that multi-tasking idea interrupting his ability to perform as well as Albanese might.