Can you charge a car battery with a 12v battery charger and use the battery at the same time?

It should work very well with a few limitations.

If you try to start your car from a flat battery connected to a battery charger, then this would in general mean that you try to start your car with the charger unit.

A charger unit can in most cases not deliver more than 6-10 Ampere at most.

To start the car, you would need 50-100 Ampere, something a normal charger can not give.

Likewise, if you turn on the headlights, these drain approx 55-75 Watt each. Depending on connected charger, the headlights could drain more power than the charger is capable of, and you would still be running the battery down, just not as fast as you would without the charger attached.

If you play music in the car whilst being in your own garage doing repairs or cleaning or any other work, then it is a good idea to attach a charger to the battery first. This would ensure max capacity when you go out for a drive since you will basically only use the power from the charger when playing music.