Can I charge a 12V 71Ah Car Battery with a 12V 12Ah - 48Ah Charger? When I try the charger says the battery is full, but the charge indicator on the battery says empty.
You can perform maintenance or "float" charges with underpowered battery chargers, but performing bulk charges (i.e. if the battery is completely drained) can burn out microcontroller in the smaller-powered "smart" chargers, or worse, cause the release of Hydrogen gas from the battery.
In summary, you CAN charge it with a lower powered battery charger, but it's not recommended when the battery capacity is >20% over the maximum capability of the charger.
If the battery is a Sealed Lead Acid (SLA), try rocking it gently side to side and tilting it slightly (no more than 30-45 degrees) a couple of times. Do you hear any liquid? If you do hear a liquid inside, let it sit for 15-30 minutes after.
If you have a multimeter/voltmeter, attach that to the battery in the 20 Volts DC setting and see if the battery is 10.5V or over, otherwise you may need a new battery. If it's in the 13.5V range, it's fully charged. Additionally, after sloshing and letting it sit, look at the battery indicator on the top of the battery and see what it is indicating.
WARNING: Do not tilt the battery completely on it's sides or hold it upside-down!