I would offer the concept that this type of competition does exist - outside the company. The outsourcing of IT is a huge trend, at least in the US - and it appears to succeed when another group can do the work more efficiently from either a cost or quality perspective than the in-house staff. I'd also offer the idea that over time more and more IT functions will morph into non-internal solutions - for example, cloud services instead of in-house storage systems. Just like COTS products that can be widely used across companies replace custom built software when the need can be generalized.
I'll be honest that I've never heard of what you propose. I have seen cases where two groups of a company provide competing IT capabilities, which eventually leads to a grossly inefficient turf war, the end result of which is usually the elimination or amalgamation of one of the two groups, or a case were executive management has to brutally delineate the lines.
I'd posit that within a company, the idea of dividing into competing groups doing exactly the same thing has the following problems:
Corporate inefficiences are not often unique to a group - there are cases where a single department is badly managed, but if inefficiencies are widespread, another competing group will only add to the problem. I've seen plenty of inefficiencies introduced by finance, security processes, management structures, and the corporate need for consensus or massive CYA activities. A second competing group won't fix any of this.
The best case for internal IT is when there is a need for single ownership and one identified responsible entity/decision maker. This comes up in security - you can't have two groups assessing/fixing security without chaining it up to a final decision maker - or else you'll generally wind up with a kludged together, insecure system that no one can reconcile. In these cases, competition will do you more harm than good.
When one entity pays the bills, competition is rarely the most efficient option. Having two groups do the same thing differently is going to lead to a redundancy. I'd have a hard time understanding the busienss case for the redundancy, as the company could end up paying literally double for the work in the end.
When it comes to fixing internal processes while keeping them internal, instead of true competition, I've seen cases where a given up and coming group or manager is given a peice of particularly slow work. The team is tasked with speeding up that one function and given particular driving milestones. If the work is part of other teams, the work responsibility is removed from those teams, or their work is de-emphasized. If the new team can do the work better - it survives and thrives and usually takes on more work... and there's usually and upper management buy in to removing obstacles.
Once the work speed improves, there are usually reductions in the teams that couldn't get it done to start with...