The word "mint" really takes the meaning out of the word "mine". True though, minting is part of the economics included in mining.
Usually governments mint coins, but no government (or centralized entity) currently operates a legitimate network that matches up with the same properties as bitcoin.
Governments need to mine or turn to mining companies to get the raw material that makes their coins. This is what you are doing with cryptocurrency. You look for the bits that make a coin valid and then the network mints the coin.
