Basically, the miner is providing the labor of creating the coin -- it's performing work. The labor cost of that work is supposed to give the coin value. The process of mining (usually hashing) can be used for different things, both tangible and proprietary. Hashing at its basic level is just doing math and then verifying said math -- it just depends what the purpose of the math is, which I believe can vary among coin types.
Blockchain tech itself can (and is) used outside of the coin realm for other purposes, and it has room to grow into that mold. It'll be a while, but I bet it will catch on largely due to how easily scalable it is. It will show to have uses in research.
You said, "The process of mining (usually hashing) can be used for different things, both tangible and proprietary."
This is what I'm getting at. What is an example of a tangible and proprietary benefit of mining other than the "creating of coins?" For instance, is the "verifying math" related to math that is being done for any purpose other than "mining coins?"