I have always half assumed that the need to acclimatise to heat was more psychological rather than physiological.
There are definitely physiological changes, too – blood density changes, sweat rates and so on. From all I’ve read it takes about 2 weeks of exercising to adjust. That seems to match up to personal experience, as the heat here tends to rocket up mid-June (so sometime next week, probably) and the first week or so is pretty grim. That said, experience is definitely a factor, too, and of course running locally I know what clothes to wear, where there’s likely to be water, the amount of shade etc.
Full disclaimer: I’m not a doctor, just a runner that needs to take this into account every year.