Thanks everyone for your help on this. After replacing bulbs and getting intermittent success in getting the light to go off, it turned out that one of the two metal socket rails in the plastic taillight housing was the culprit.
Pressing down on the rail caused the brake light bulb to change intensity, so some kind of short or connection problem exists. I didn't have time to remove the rail to further investigate the problem, but I did bend the rail slightly to eliminate the bulb flicker.

No more bulb indicator light staying on!! Been staring at that for nearly eight months now.

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2009 ML350 (106K) - Family vehicle
2001 CLK430 Cabriolet (80K) - Wife's car
2005 BMW 645CI (138K) - My daily driver
2016 Mustang (32K) - Daughter's car