I'd prefer to consider a PADI OW "certification card" a device akin to a diploma.
The document is proof that you have successfully completed a course of instruction that makes you competent to effect a scuba dive to a recommended depth.
The document also provides you a basis of education within the discipline upon which you can advance your studies and knowledge.
A "license" is generally regarded as a document required by a governmental agency to perform certain tasks or jobs.
One can scuba dive without a certification without fear of arrest from law enforcement agencies (in most places in the world) unlike driving. Just a comparison.
the K
The Kraken is right. The C-card (from any agency) is not a license at all. There is no state or federal law in the USA that requires you to be certified to dive. A C-card's validity exists solely within the dive industry. Dive shops and dive ops require them before selling you air or taking you out on a boat. As far as the government is concerned, nothing has changed since the first SCUBA equipment was sold in this country back in the 1950s.
It is quite possible, and entirely legal, for a person to teach himself to dive, and continue to dive, outside of the realm of any agency. All one has to do is study the wealth of knowledge available on the science of SCUBA, work carefully through learning the basic skill set and practice. Everything you need is available online without the requirement of certification, including a SCUBA compressor to fill your tanks.
Or, you can do as I did: I was taught by an ex-Navy diver who was, himself, not certified by an agency. He taught me as he was taught. He had a compressor, filled our tanks and we went diving. I was a diver for 11 years before even considering going for a C-card and then only because he passed away and I needed the card for air fills.
So, no, a C-card is not a license or anything like a license. It's true function is three-fold: It's a CYA sort of thing for LDS and dive ops; an attempt to keep the government out of the sport and it's a way for the agencies to extort money from divers.