We have a two party system (we pretend otherwise).
Not really – parties are coalitions internally two. Most likely we have at least six parties: the left and right wings of Labour and Tories, the completely batshit right wing of the Tory party, and the Lib Dems (which is more of a smear than a bloc). Blair's impact was so strong because he had a Blairite majority so that he didn't have to rely on the left wing of the Labour party.
And then of course there's the 4 party systems in Wales, NI, and Scotland.