So first, there things known as dashpots, for which the force really is proportional to velocity. These are the things that intended to be represented by the damping term in the classical mass-spring-damper systems that you find in the textbooks.
But there is a much bigger picture idea here. The mass-spring-damper systems in textbook are just mathematical models. The important thing to realize about models is this: All models are wrong, some models are useful. We use the mass-spring-damper model not because it is exactly correct for real physical systems, but because it is simple to understand and is close enough for many purposes.
Specifically regarding damping, there are some systems for which the damping really is proportional to velocity squared (or maybe some other relation such as dry friction damping, aka coloumb frction, for which the force is proportional to the sign of the velocity). But including a velocity squared term can make the resulting equations much more difficult to solve. e.g. it could go from 30s to solve the equation to 3 hours to solve the equation. If we absolutely must know the exact answer to within 0.1%, we will spend that 3 hours. But if we only want an approximate answer to within, say, 10%, we will take the easier route and just model the damping as if it was a dashpot. We should remember that neither model is exactly correct, both of them are wrong, just one is closer to the truth than the other. We pick the one that is most useful for the given situation.