Incumbents count as competition as long as they are in the same market. If we just look at NGO:s the list of competitors becomes shorter, of course. Falcon 9 launches (8 in 2016) were 9.5 % of the total global launches in 2016 (85) [0]. When appraising the future of spaceflight, It would be quite misleading to ignore e.g. Indian and Chinese efforts in this sphere, government funded or not. This is not to say I'm not silently cheering each time a Space X launch succeeds.
Most of the launches are simple not competitive. SpaceX can not bid on Government contracts in China.
The actual commercial market for launches is getting taken over by SpaceX. The only thing that holding them back is launch cadence and their government contract requiring them to make launches (COTS, Commercial Crew, USAF and so on).
They are building more launch sites as well, so they are really just pulling away from the competition in that area.
[0] https://en.wikipedia.org/wiki/2016_in_spaceflight