With humans, eye contact can mean different things depending on one’s culture.
In the U.S. and most of Europe, eye contact signifies genuine interest in another person. It can also be flirtatious and a signal that the person making eye contact is confident and bold.
In other cultures eye contact can be seen as a challenge or an offense. These are just some of the meanings in different cultures involving eye contact.
Feral cats and dogs you are not familiar with can view eye contact as dominance or a threat and a signal to challenge this stranger.
When meeting a strange dog it’s best to stand still and avoid looking directly at him. At the same time be aware of what the dog is doing. If you run, the dog may view you as prey and will take up the chase. In a strong voice tell the dog to stay or sit or go away.
Studies have shown that because dogs have been domesticated for so long, many of them have acquired human habits. Making eye contact can be an expression of how much their trust in us has grown.
My own dogs do look directly at me and will hold eye contact for a time. I’ve found that squeezing my eyes and/or blinking and whispering “I love you” or anything in a gentle voice puts them at ease.
Horses need to be looked at with a “soft eye” or gentle look. Horses are prey animals and if you are very focused and make direct eye contact, they may think you are a predator and will shy away.
In general, most animals view direct eye contact or staring as a threat. Prey animals will run away and predators may attack.