"In March 2015, Square introduced Square Cash for businesses, which includes the ability for individuals, organizations, and business owners to use a unique username to send and receive money, known as a $cashtag"
But for some reason they neglect to point out the pre-for-business component. It actually dates back to 2013:
https://squareup.com/news/introducing-cashtags
I stand corrected, though, because as you note, Venmo predates it by several years. I guess I just had no reason to switch from PayPal until Square Cash came along in 2013 and was even more frictionless. But ultimately, these things are clearly down to network effect.
Oddly, all I can find about the PayPal mobile app is that it was "re-designed" in 2013. Of course, I was using PayPal on a Palm handheld back in ~2000....