Tiredness after work is real, but exhaustion from caring for two little kids all day is a completely different level of burnout. Here's the thing: you're both tired, but your tiredness doesn't end when he walks through the door. Night wakings, constant attention to the kids, no breaks - it all adds up and leads to burnout way faster than work does.
Can things change? Yeah, but only if he wants to actually understand it and agrees to change something. Try talking not with requests like "help with the dishes," but with specific offers: take the kids on the weekend from 10 to 2, and I can just sleep or have some time alone. Or agree that three times a week after work he doesn't watch TV, and you guys go for a walk with the kids together - that's both family time and helping you out. If talking doesn't help and he keeps thinking that only you work in the house while his role ends with bringing home a paycheck, maybe it's worth considering a conversation with a therapist together. Sometimes people don't see the problem until a professional points it out.
Is it his character or a habit? More like a habit. And habits change slower than we'd like, but they definitely do change if there's the will to do it.