How would inter-operability differ from sharding? Is the difference that sharding only works with a specific blockchain technology, and inter-operability could connect multiple, technologically different, blockchains?
Think of it like TCP/IP today. In your own network, you can use whatever you want, let's say Zigbee. Once I want to interop with other networks though, I'll have to go to TCP/IP or similar.
There is a reason that you can't have everything on one blockchain.
And that is because blockchain scales in an N squared manner.
Every transaction that happens, has to be seen by EVERY node on the network.
The value of the network does go up the more people use it, and therefore, there are more transactions per user. However, the amount of transactions that each user produces will probably not scale linearly with the network size.