The Chang’e 3 moon lander was not named for “the Chinese goddess of the Moon.” Chang’e (or Chang-o/Heng’o, in the older Anglicization styles) is a woman who either fled to the Moon after stealing either the Peaches of Immortality or an immortality pill made from the peaches, or a goddess trying to regain her immortality who ate too much of a magic immortality pill and thus floated uncontrollably up into heaven until she was stopped by the Moon. Either way, she was unable to leave again. Some have her fleeing her husband, the archer Houyi who saved the solar system by shooting down nine out ten belligerent suns, and some have her longing for him all this time.
So she’s a goddess who lives on the Moon, but she’s also a prisoner, a castaway, or an exile. She doesn’t personify the Moon or control it. She is worshipped mostly for her presumed ability to provide her worshippers with beauty and grace, as she is said to have been the greatest dancer among all the Chinese pantheon; she is also associated with immortality.
More seriously, the Yutu lunar rover, aka Jade Rabbit, is also not named for any “pet” of Chang’e. The Rabbit on the Moon is what Chinese and Japanese people see instead of the Man in the Moon. His “jade” color is white jade. In the Japanese tradition, the Moon Rabbit makes mochi (sweet rice treats), but Chinese tradition has him making various kinds of magical elixirs. One tradition says that he’s been working all this time on making a pill to let Chang’e fly home. Anyway, he lived on the Moon before Chang’e ever came along, and he probably was originally a moon god — although now he’s also considered to be just a deity who lives on the Moon.
(Of course, rabbits and the Moon have strong associations all over the world, probably because rabbits sometimes go out on moonlit nights to eat and play, whereas most daytime animals don’t. There’s also an association of ideas between the Moon and rabbits’ obvious high fertility rates — and of course, some other people besides the Chinese see rabbits on the Moon’s face.)
Some people see the Man in the Moon as carrying a bundle of sticks, and this viewpoint is represented by the woodcutter Wu Gang, who is also immortal and trapped on the Moon. He can escape if he can cut down an ever-growing cassia tree, which either represents his own life or all mortal life on Earth, but he is never able to chop faster than the tree can grow. Some people also saw a toad on the Moon, but there doesn’t seem to be much on the Internet about that.
(Insert obligatory MLP reference to the imprisoned Mare in the Moon.)
Characters inspired by these folks are always showing up in anime and manga. Probably the best example is Sailor Moon, whose personal name (Usagi) means Rabbit. In the Space Brothers anime, Hibito (who becomes the first Japanese man on the Moon) is given the nickname Hibbit for his habit of hopping around, when in low gravity.
Anyway… my point is that the media doesn’t do a lot of research on this stuff, so you’ll have to look it up for yourself.