I think a lot of the replies you're getting OP, skip over the fact that boys are fundamentally different to girls in their biology. Boisterous play shouldn't be discouraged - even aggressive play shouldn't be discouraged. This is how small children process & learn. The key part is that aggression towards themselves or others shouldn't be tolerated.
I do think the example you gave of a bully at home, a puppy outside, was ill chosen. Home is a child's safe place, of course they're going to be more comfortable acting out of exploring different types of play there. It sounds like you have a very normal little boy.
I'm still in the process of trying to raise a well adjusted boy, so by no means an expert. Things I've researched and am trying though are below - obviously they'll work for girls too!
I do not equate hitting or fighting in play with signs of an aggressive personality. I do not give out at this or try redirect it, unless it's harming himself or others. I don't buy weapon toys, but little boys will be creative in making swords, etc.,
I give him lots of opportunities to get his energy out, and focus on boisterous / sensory play. Boys love and need rough and tumble. Giving safe opportunities helps. Lots of jumping on the couch / throwing him on the cushions safely, tossing him in the air, spinning, bike rides, jumping etc., If there is play fighting, I make sure he knows how important consent is, and the boundaries around when this type of playing starts and finishes.
I encourage confidence and independence in his care tasks. E.g. he's learning to cook from a young age, and pick up after himself, in a fun age appropriate way.
When he's having big feelings, I help him recognize what they are by naming them. "I think you're feeling sad your toy broke."
I spend a lot of time helping him co-regulate during big feelings too.
I reward kindness and good behaviour. I model the way he should be treated, and should treat others. Very importantly so does his dad.
I make sure the TV be gets to watch is appropriate. E.g. we don't watch Bad Dinosaurs on Netflix because the characters are frequently shoving and hitting.
My expectations are also mapped to his age. My son has toy dolls and non typical gender toys, but all he wants to play with are cars and dinosaurs. He is little, he doesn't need to practice being nurturing any more than our daughters do - he needs to be the one nurtured. I let him lead the way, and focus on loving him. He will have tantrums, and instead of seeing these as a sign of potential future bad behaviour, I focus on what he's learning from these in terms of regulation.