That depends on whether you are Greece (call it gyros) or Turkish (call it kebap) or whether you want to risk a fight (calling it gyros in front of Turkish people or vice versa).
With a Döner plate usually mixed salad is served and the yoghurt dressing would be on top of the Kebap meat. With a Gyros plate usually Tsaziki and coleslaw are served.
The things on top of the pudding look like decorative fake coffee beans made out of coffee flavoured chocolate. I'd be willing to take bets that the pudding is also coffee flavoured to stick with the theme.
You need yoghurt, the heavy stuff with 10% fat, olive oil, garlic, cucumber, pepper, and salt. Nothing else. No, no dill, no mint (WTF?), no nothing.
Julienne the cucumber. Very fine is better than fine as long as you're not producing mush. Salt it, let it stand for 10 minutes, then squeeze dry, toss the water. Add yoghurt, should be about two to three parts of yoghurt for one part cucumbers, by volume, don't sweat it. Take about a clove for 500g of yoghurt (that's a clove, not a bulb, yes it's quite little, but it's raw and it's going to infuse), surgically remove the sprout (that's where the nasty stuff is in garlic), chop finely. I said chop, not squeeze, yes it makes a difference. Add with pepper and salt and some olive oil, put in the fridge for at least one hour better a day, well covered (closed container is good, cling film if you have to), mix again and do final taste and consistency adjustment with pepper, salt and olive oil. Pepper should be subtle AF, supporting the garlic, not supplanting it.
...it's absolutely fine to do other yoghurt sauces and in fact in Germany you'll see three or four at any Döner shop, but don't call the non-tsasiki tsatsiki, please. If you want a herb sauce, call it herb sauce. There's no herbs in tsatsiki. (Sauces differ regionally in Germany -- there's always going to be tsatsiki, around here you also generally get curry, hot or mild, as well as cocktail sauce (no, not mayo based, it's still yoghurt)).
Oh I couldn't call any non tsatsiki sauce tsatsiki, since I actually hate it (sorry Greeks or Turks), that would actually be an insult to the sauce. :P