You're making an argument based off of PF, which is largely a reskinned 3.5E? And your argument for why your dwarf feels dwarfy despite having halfling equivalent stats is because you've given him an axe that he's really not going to be any good at using?
No, I believe the argument is the dwarf feels like a dwarf thanks to unique feats and features (which happen to include proficiency with iconic weapon groups) despite not receiving an arbitrary penalty to their spellcasting ability as they would in core 5e.
Also, you're mistaking Pathfinder 1e and 2e there which are vastly - and I mean vastly - different games with different design philosophies.
