"post offices can provide many of the same services OF banks"
The word "same" takes the preposition "as" when used in this sense. "the same services of banks" is wrong.
The following are correct:
- post offices can provide many of the same services as banks.
- post offices can provide many of the services of banks.
- post offices can provide many of the services provided by banks.
- post offices can provide many of the services that banks do.