Passport fits perfectly in a thin nylon neck pouch and is way more secure than a regular wallet.

I consider a neck pouch the most secure, followed by leaving valuables at the hotel/airbnb, followed by carrying a regular wallet or purse.

Incidentally, in my experience not every place has a room safe like you might be imagining so you may want to give some thought to how to carry valuables with you in case that is sometimes your best option.