In the novel Cold Equations: The Persistence of Memory, Dr Soong writes his software for gambling machines which monitors the bio-signs of the participants and manipulates the game depending on the read-outs to make the maximum profit. It strikes me as somewhat unlikely that Starfleet would allow LCARS to be used for gambling ventures...
In the same book, Dr Soong also details how he worked for Starfleet writing his own security software for them (and left back doors for himself).
All the software which he wrote for the Soong-type androids is evidently compatible with LCARS, possibly built on it, but is his own (and Juliana's) software.
Whether you want to call this software 'Soongian' brand software or not, it's up to you, but there definitely are people who independently write software that isn't built on independent software that is not LCARS.
The other aspect to remember is that we are only focusing on the Federation! What about all the other computing systems that the other governments use. What about the Ferengi? In the extreme capitalist society they are, there probably are competing proprietary software brands there. We also know for a fact that the Klingons, Cardassians and Romulans all use different software systems than the Federation, meaning there is also potential there for brands to creep in there.