This whole 'club having a philosophy' is a bit over-egged. Of course it figures that a team doing reasonably well playing one style will continue to do so if a new manager comes in with the same thoughts, compared to it not going so well if a manager tries to play a different style.
Check out Moyes and Van Gaal as evidence compared to what Ferguson's Man Utd was. Its not rocket science that all of those players don't fit the mould required of them and so don't produce results. We've chopped and changed over the years - from Evans, to Houllier, to Benitez, to Hodgson, to Dalglish, to Rodgers. The only real link in transition was Houllier to Benitez insofar as they both liked counter attack. The rest was a real mishmash.
Its our old 'Bootroom' approach, i.e. a method that promotes continuity. The key is that its built upon success.