“Drink stabilizer” describes the goal (reduce tipping/wobble), not one single product. Here are the main types and when each tends to work best.
A stabilizing sleeve typically increases the footprint at the bottom and helps the container sit more confidently on soft or uneven surfaces.
These add mass low to the ground. They can help on hard surfaces where friction is good and the goal is resisting bumps rather than deforming surfaces.
Examples include couch cup holders that “park” a drink in one place. Great when you want a designated slot, less ideal when you want to set a drink down anywhere. See couch cup holder vs stabilizer.
Mats, grippy coasters, or adhesives increase friction under the container. They can reduce sliding, but may not help on soft surfaces where the container still tilts. See coaster vs stabilizer.