Sadness, like happiness, can cause tears for some simple reasons. If the emotion is strong enough, it will release a special hormone, which will make people "cry". This reaction inevitably makes the bystander subconsciously think that the victim's system is losing fluid: he (she) is crying.
Crying is a signal to express "I am in a bad situation and need help". Whether tears exist or not will affect people's judgment of expression. For example, if there are no tears in a crying picture, it will be more difficult for you to judge whether this person is sad or not.
This signal seems to be unique to human beings. Because the apes closest to human beings don't cry when they beg for help. For animals, crying is an effective way to communicate information. First of all, this animal needs to have a complex brain, communicate frequently with its companions, and use facial expressions and sounds as important signal sources in communication. I think the primates and carnivores that meet these conditions best, but the "most human-like" ones don't cry.
For animals other than humans, the closest thing to crying may be an interesting story recorded by Darwin in "Expression of Emotion between Humans and Animals": an elephant cries when it makes a loud cry like a horn, which seems to be just a muscle action when it crows and a by-product of oppressing the lacrimal gland.
There are still differences between people and animals. People express their feelings by crying, but it is difficult for animals to express their feelings.