Chelsea don't need to fire Pochettino after just 6 games, but they need to push him to get his team better. I don't think Pochettino managed to make the team's dressing room any more lively regardless of the outcome of the match, but tensions became more likely. Pochettino has to start from small things and improve them, that includes improving the atmosphere in his team's dressing room for the better.
Liverpool also have many players whose quality is almost the same as that of Chelsea, but Klopp is here to improve it and get his players into the best shape. He turned his team into a competitive team even though Klopp has not won as many titles as Manchester City. Pochettino must learn from Klopp and Arteta instead of being a coach who tends to stay silent on the sidelines.
Chelsea's performance has again achieved poor results and it is very natural that Pochettino is in the sharp spotlight and has received a lot of criticism. because from six matches Chelsea are still slumped in the middle of the standings, they are ranked fourteenth with a collection of five points.
Chelsea only won one win from six matches in the EPL, the only victory Chelsea achieved was against promoted team Luton Town. Of course, these results are very unsatisfactory. Moreover, Chelsea spent big in the transfer market last summer. I think if Pochettino can't improve this situation, Chelsea might look for a new coach if Pochettino continues to get bad results in the next match. So it cannot be compared to Klopp at Liverpool, he was never as bad as Chelsea, he can learn from mistakes.