Is "purchased" more formal than "bought?"

Macmillan tags purchase as formal, while their definition of buy in the sense of of "to get something by paying money for it" has no such qualifier.

The phrase bought it is a colloquial or informal way of saying killed. But bought on its own is, as the linked definition says, simply "the past tense and present participle of buy."

EDIT: M-W lists boughten as a dialectical form of bought, and provides this delightful example:

my red sled, and my boughten wagon --W.A. White

Bought isn't generally considered colloquial, but purchased certainly is more formal. There's no reason it would be considered improper regarding recent purchases in particular, unless for some reason any shred of informality is forbidden.

It may be useful to look at the most common collocations of each in British English:

PURCHASE PRICE
PURCHASE AGREEMENT
PURCHASE ORDER
PURCHASE BEHAVIOUR
PURCHASE GOODS
PURCHASE MONEY
PURCHASE SCHEMES
PURCHASE SHARES
PURCHASE DECISIONS
PURCHASE AGREEMENTS
PURCHASE TAX
PURCHASE DECISION
PURCHASE ORDERS
PURCHASE GRANT
PURCHASE LAND
PURCHASE TERMS
PURCHASE CONTRACT
PURCHASE CONTRACTS
PURCHASE NOTICE
PURCHASE LEDGER
PURCHASE POWERS
PURCHASE CONSIDERATION
PURCHASE MOTIVATION
PURCHASE SCHEME
PURCHASE SERVICES
PURCHASE ARMS
PURCHASE FUND
PURCHASE GRANTS
PURCHASE MONIES

BUY SHARES
BUY FOOD
BUY THINGS
BUY GOODS
BUY TIME
BUY TICKETS
BUY LAND
BUY EQUIPMENT
BUY CLOTHES
BUY BOOKS
BUY PROPERTY
BUY PRODUCTS
BUY HOUSES
BUY PLAYERS
BUY DECISION
BUY DRINKS
BUY PRESENTS
BUY STUFF
BUY UNITS
BUY SERVICES
BUY BREAD
BUY DRUGS
BUY FISH
BUY STOCK
BUY COMPUTER
BUY CONDOMS
BUY HOMES
BUY RECORDS
BUY CIGARETTES

Most of the collocations for purchase have to do with business interactions, where communications and descriptions are usually formal. Most of the collocations for buy are for daily goods, which would usually be mentioned in informal conversation. Even if neither is usually explicitly identified as formal or informal, the usage pattern shows it.