the Phrase "stop being a pussy" is a slang term describing the actions and whining sounds that cats can create. These utterances of the feline members can be found to be annoying as is the constant actions of rubbing its body against yours.
In this context, being a pussy is to in fact to be whining and annoying.
"I don't like how everyone keeps on ignoring me!", says John; "stop being a pussy", comes the soft lull within the crowd