It's a tap card originally for public transport, but now widely used in stores, parking lots, post offices, fast food outlets, etc. throughout Hong Kong. It's very popular, and just about everyone in Hong Kong uses it every day.
It's anonymous in the sense that no ID's required to buy one.
But the cards are all numbered, of course, and if the Authorities know the number blind stamped on your Octopus card, then they can -- in theory at least -- match it with the payment record kept by Octopus. (Although Octopus is run as an independent, private venture, the HK Government effectively controls it with about a 60 percent indirect shareholding.) And with a maximum loaded value of HK$1000 (US$125) it's not really a high-value payment option.