Just curious, is that still true if your model of daily returns isn't gaussian? If prices have intermittent shocks (Ornstein-Uhlenbeck, etc) is the daily vol much higher?

Vol is a parameter in a model as well as an observed statistic. Naturally you can have jump models as well that have other parameters, but normally when we talk about it we mean the observed stat, with a standard normal implied when discussing it. Weirdly I've not often had a conversation where someone talks about alternative models, even though people do use them.