That implies the spare tire has a different circumference than the other tires. Something may not be original because manufacturers make sure the spare is the right size to work on ABS-equipped cars.
It's also possible a wheel speed sensor got damaged by the flat tire. You didn't say what caused the flat, which tire, or whether you were driving when it went flat. All of that information can be helpful to know. If a front tire went flat while you were driving, the flopping sidewall may have torn the wires going to the wheel speed sensor.
It is also possible the diagnostic fault code that set in the computer is a "latching" code meaning it will not erase on its own after you stop and restart the engine. Your mechanic may have to connect a scanner to read and erase the code to make the warning light turn off. As long as the light is on, the computer has the anti-lock function turned off.