Tail numbers were the Bureau Numbers assigned at the acceptance of the aircraft at the manufacturer. Squadrons had a unit assigned number, and paint scheme. The scheme would allow a visual id of what squadron the a/c belonged to, and at the same time, what carrier, if it was sea based. The unit number was generally a 1 or 2 digit number, but today, it has evolved into a 3 digit number. My uncle was an aircraft commander in WW2 with VP-74. His assigned aircraft was P4. In they're case the P was to designate a combat aircraft over a transport version.
On the radio, they were identified by the assigned squadron/flight call sign and a specific number within the squadron. For example, one squadron might be assigned the call sign "Rattler" for a period of time, or even a single mission. Pilots within "Rattler" would identify as "Rattler xx" based on their squadron assigned number.
A good example is our company test pilots. On the radio, they are known as "Rocket xx" for certain types of flights.