It's pretty common with people I know born from 1980 and beyond. Of course, it gets more common the closer their year of birth is to the aughts. But I guess you are right, everyone doing it today was under 20 at some point.
As someone born in the late 80s I've only heard about it from the press. I really sometimes omit periods in one-sentence text messages, but sometimes I don't, and it wouldn't come to my mind to ascribe any semantics to this.
My personal nitpick is people writing a stream of one-sentence messages without punctuation instead of a longer structured message - that definitely carries the connotation of unstructured thinking to me - but it doesn't have much to do with full stops per se.
Unstructured messages in a serious place is annoying, yes, but in a water cooler area it can set a more relaxed tone. It's a cultural thing and I'd never force someone to endure my style of communication in a place where it's not welcome.
