No, Tesla will NOT have a monopoly on LI batteries. There are large suppliers coming online in China, and there are several other smaller suppliers sourcing similar batteries.
But what Tesla has done with the Gigafactory is to optimize the production, packaging, and management of the batteries to allow them to have higher capacities, faster charging, and longer life than other existing battery suppliers. They have also minimized the amount of cobalt needed, which is actually a tighter constraint than lithium for similar battery makers.
There are others working on solid-state batteries and other technologies, but none of them are scalable yet to the required levels.