It wreaks of bitterness but his adored mother referred to doing her duty, producing an heir and a spare, whether other people referred to him as a spare before her I wouldn't know so I couldn't say it came from her in the first place, but it was a phrase she used publicly.
Tbh I think the concept of heir and a spare is common among the landed gentry. Of course in the case of Diana and her brother Charles, she felt unwanted as there was pressure for her mother to produce a male heir. The opportunity to produce a spare never arose in their case, but the concept would not have been unfamiliar in those circles.
I have a wealthy b-I-L (massive odious snob) whose older brother inherited a substantial estate and being the younger one he got a holiday home on the estate and I suppose some inherited wealth plus an easy job for life via connections. He would defend primogeniture to the hilt but none the less I think he's a bit bitter at not getting the full whack.
Younger siblings up and down the country miss out on the lottery win. Most get on with it. In Harry's case it meant he could have enjoyed many of the privileges and wealth whilst not having the onerous responsibilities. It wasn't until a certain person came on the scene that it ever occurred to me he might feel bitter about his position in the pecking order rather than grateful.
It's honestly a mystery to me. I've no idea why he should feel resentful about it but it seems he does.