Yes — and this trips a lot of people up. Your car is just another load on the house, the same as a kettle or an immersion. Solar generation merges with grid power on the same supply, so whatever's running in the house is fed by that combined mix at that moment. When your panels are generating more than the house is using, that surplus is what feeds your appliances first, and anything left over is what exports to the grid. So if you plug in a plain three-pin granny charger while you're exporting, the car will be drawing on that solar surplus, not magically pulling everything from the grid. A dumb charger doesn't 'see' the grid and decide to import — it simply takes its power from the live supply, which is solar-first whenever the sun's doing the work.
The real difference a smart charger (Zappi-style) makes isn't where the electricity comes from — it's control. A granny charger pulls a fixed ~1.4–2.4 kW the whole time it's plugged in, so on a cloudy spell or once the sun drops it'll keep going and start topping up from the grid because it can't throttle itself down. A smart charger can be set to charge from surplus only, ramping up and down to match what your panels are actually exporting, so you import little or nothing. Same physics for a plug-in hybrid (PHEV) — it's house load either way.
If you've no smart charger yet, the practical move owners use is to charge on bright days when you can keep an eye on it, or just charge the car overnight on a cheap EV/night rate (often around 2–5am) and let your daytime surplus export for the Clean Export Guarantee payment instead. The best economics usually come from a smart charger plus a night tariff, but a granny charger absolutely uses your solar when you're generating — it just won't be as tidy about avoiding grid top-ups.