For AE, the various terms overlap somewhat, but Hendrik is absolutely right that 'purse' in any form is only for women. (Though, Hendrik, I think your friend probably didn't say 'wear' but 'carry' or 'use' or 'have.' 'Wear' is for things you don't have to hold on to: clothing, shoes, hats, wristwatches, jewelry.) (-:
wallet = flat folded squarish leather container for bills (= paper money), credit cards, & receipts (no coins)
billfold = (1) wallet; (2) woman's larger wallet including a coin compartment and checkbook
purse = woman's handbag
coin purse = small purse for coins only
pocketbook = (1) traditional-style leather purse with handles and snap closure (think 1950s-60s) (2) woman's billfold
My grandmother's generation carried coin purses inside their regular purses; bills could be folded inside the coin purse, and a checkbook could be carried separately in the purse. Now most women have billfolds, which come in two shapes: checkbook-style (long) and wallet-style (short). Some businesswomen now use a sort of combination purse/briefcase with built-in or detachable billfold features and more compartments for things like cell phones.
Most American men carry wallets in their back pants (= trousers) pocket. Coins are kept loose with keys and other small objects in the front pants pocket. Checkbooks (if any), passports, airline tickets, etc. are carried in the inside breast pocket of the suit jacket. Men who need access to more papers carry a briefcase and/or a laptop computer.
Men's leather document containers that look like a flat clutch purse with a corner strap are seldom used in the U.S. because they look so feminine, like a woman's clutch purse. If we had to call them something, it might be a man's pocketbook, but the very idea is probably a shock to most Americans' systems. (-: