This doesn't sound right to me that the static voltage (voltage of the battery at rest, when it has no drain on it and is not in use), and the charging voltage (voltage at the battery with the car running and voltage from the alternator is being sent to the battery) are the same.
The correct procedure for testing these are:
1) Drive the car around for a bit then come home. Blip your headlights on for a few seconds to get rid of the surface charge on your battery, then let it set for several hours. Take the multimeter negative and positive ends and attach them to the corresponding negative and positive ends on the battery terminals. A fully charged battery will have 12.7 volts. If you have more than this, you have a surface charge still and are not getting a try reading of battery state yet. If you still read 12.3 volts, either A) your car has a parasitic drain somewhere and is using power while the car is off, and your battery is in a point in it's life cycle that it is actually only retaining 70% of charge. (state of charge) If it's much lower than this, you should probably consider replacing the battery with a NEW unit. (also, a battery could have a high fvoltage, and lower CCA, which stands for Cold Cranking Amps, this is only measurable by an old-school load tester which is sometimes referred to a "toaster", or a current age digital analysis system that costs hundreds to thousands of dollars. If you are concerned about the state of your battery at all, or just for piece of mind, take your battery to Autozone or Advance AutoParts (I recommend a Advance because they use a high quality Midtronics tester with printer) and will test your battery for free)
2) For testing the alternator charging voltage, keep your multimeter negative and positive ends still on the same corresponding points with the battery. Then, turn the car on, and notice the spike in voltage. Then, turn on the defrost and heat on HIGH, and headlights on, and notice if the charging voltage maintains. This is important because this is a load test and will test that the battery and alternator are working functioning under load. For this car, you should definitely have over 13 volts, probably closer to 13.5 or 14. This completely depends on the quality and condition of the alternator and that the belt on the alternator is not slipping. If your charging voltage is not within range, then replace your alternator. I recommend buying a NEW and not a rebuilt one, much less headaches down the road.
Also, stores like Autozone and Advance Auto Parts will also bench test an alternator in-store for free. Their tests may not always be an exact science, but will give you a much better idea for sure of how much voltage/amperage it's putting out.
Good luck and hope this helps.