Men are typically not told to 'calm down' nor are they accused of being 'emotional'. (The only emotion they're usually allowed is anger.)

Tellig a woman to calm down doesn't mean "I can relate to men better", so much as "Look how silly and feminine she's being and look how I'm correcting her"; it means she's employing a sexist response (like a guy would) to demean a woman.

Her policy may sound logical but that has nothig to do with the comment to the other MP. She should have replied with proof that her policy was logical. Tone-policing her was not only demeaning, it was deflective (which is a topic I've also written about here.)

The pick-me element comes in because by treating another woman like she did, highlighting the emotion (the need to calm down), she's trying to show that she (the woman who's not like other women) doesn't get ruffled and won't tolerate it in others. (Not that the other MP was ruffled at all.)