I can't speak toward tarantulas, but some jumping spiders definitely recognize, acknowledge and seemingly even appreciate individuals. Honestly, they sometimes look excited to see you as a dog would -- though that's likely just a basic recognition that you act as their food delivery service. They even appear to understand the concept (if not reasoning) behind their enclosure.

Some folks who spend a lot of time with their jumping spiders will actually leave the enclosure door open, as the spider will happily stay there unless it lacks water or food for a long period of time, and typically only leaves when its human stops by. It will come to the door when you get close, jump on your hand and hang out very calmly (some people even go on walks with them), then gladly jump back into the enclosure when you're done and patiently wait for you to bring it food.