Who did the tax collection? Did the exchange collect it automatically, or did your buddy do the filing? If the former, the exchange is sketchy; if the latter, your friend messed up.
The exchanges don't collect taxes. Tax office probably requested information from the exchange and the exchange complied, and the tax office calculated the total amount withdrawn as taxable. The idea that an exchange would be calculating taxes is ridiculous.
If a sale is reported to the IRS and the exchange doesn't know (or doesn't report) the cost basis of the coins then the person who sold the Bitcoin is responsible for reporting their cost basis. If they didn't report a cost basis then the IRS is going to come back and assume it's zero.
He did the filing with the tax form that company sent. I think the exchange he transferred from was sketchy or not confirmed. In that case, when you sell, they show the full price.