The reason behind Pochettino's recent demeanour is obvious - he feels that the team have achieved more than they ever had the right to in light of the Wembley situation, lack of signings and injuries.
The team have been running on will power and determination for 6 months, and he knows that that is not a situation that can be maintained indefinitely.
We know that he is close enough personally to Levy for them to have already discussed this. I'd highly doubt if he were using press conferences as a platform to put pressure on the club. His spiky and mercurial soundbites are usually in response to Journalists who want to exaggerate the expectations placed upon the players.
Levy is not dumb - he knows Pochettino is a father figure to half the team, and losing him would disrupt the team in a way that would take years to rebuild.
Regardless of our next game, I would be astonished if we don't see massive investment in players this summer. Liverpool lost the final last year and responded by buying a world record goalkeeper, and several players for squad depth. I would also be astonished if the same culture of last minute purchases and holding on to players for the best possible sell on price continued. Thats the new way that Pochettino was referring to.