Various governments have touted the concept of a "third banking force", going back about 30 years. It sounds good, and gets a bit of positive press for the politicans involved.
There's been suggestion of merging ICC, ACC, PTSB and An Post, in times past... More recently, Permo, KBC and UB. All pie in the sky.
The closest that Ireland has ever got, to ever having a third banking force, was while Ulster Bank was committed to Ireland - and that wasn't anything to do with the Irish Government.
Going forward, we'll see the Permo try to press for equal recognition with the two pillar banks, and get some uplift as a result of Ulster Bank exiting the market. That's more to do with the Permo's own struggle to justify its existence, than it being a true force in Banking though. Its still going to be a a very small player, compared with AIB and BoI.
When all is said and done, the reason that Ireland doesn't have a more competitive banking market is simple - Irish people don't move away from AIB and BoI, in significant numbers.