What makes humans different from other creatures on earth is our ability to reason, laugh and cry. Not fear or flight response. Tears are unique to humans only....however primates other than humans do appear to cry and laugh through vocalizatoons but lack the ability to sob uncontrollably like a human. Other primates also exibit some reasoning functions based on their language with some thought like processes. Although we are distinctly different in that no scientist has ever found any other species with tears they have confirmed the presence of types emotions, especially in mated or bonded pairs, where one is injured or hurt and the other wails and moans in distress and emotional pain, as in apes and monkeys. It has been documented that baby apes and monkeys will wail when there mother dies, without tears.
Language is essential to communicating thoughts and emotions, and different species communicate on different levels based on their language. Some species can communicate distress but may not actually feel it. It isn't really safe to compare the emotion of a human to that of another species (we are after all the only human species), because we project our feelings onto them, which does not apply.
Because we are unique, we have empathy (another unique human trait) and that makes us more understanding of our pets and other creatures needs and language. We sense their state of being at times even better than they themselves can or do.
In essence it becomes a play on words.....what we know as humans doesn't fit anywhere else, so some say no other creatures have them. If define emotions as a range such as fear, comfort, peace, affection, etc etc. then it can apply to many species.