It means the formality level of text. I think that person was saying that within their family chats, they perceive that using capitals and punctuation makes the messages feel unnecessarily/inappropriately formal.

Yes exactly.

Register of speak I suppose ? Formal, vulgar, technical, casual... I might be wrong English is not my first language. But that would fits nicely. I understand that as going from casual to formal, for instance.

“Raise the bar” would be the colloquially used term around here.