After reading multiple articles, excerpts, and statistics I can confidently confirm that this claim is true. Just as humans, animals feel strong emotions of fear, grief, happiness, and excitement just as intensely. They also depend on their environment and those around them to determine their emotions. Animals aren't as easy to study, however.
A study conducted at the University of Rennes in France, Martine Hausberger used horses to study how animals retain emotion. Using brain waves that portray emotions, she found similar emotions found in humans. As humans, we produce waves called theta waves, which are thought to reflect well-being and calmness. She also found same exact waves, theta waves, when horses were out of their stalls, roaming outdoors and gazing as they pleased. In contrast, she found gamma waves in horses when they were locked in solo stalls, which are associated with anxiety and stress (Katsnelson).
James C. Harris at John Hopkins University describes it as "'an evolutionary mechanism to maintain social cohesion.” In other words, animals that rely on a group for survival must be more sensitive to what those around them are feeling, whether they’re human or non-humans" (Harris).
Science News Today also states that animals that are held captive often show signs on anxiety. When animals are deprived of social contact, stimulation, etc., they start showing physical signs of stress such as pacing repetitively. Such responses mirror physical anxiety in humans as well, such as those with anxiety disorders.
Multiple sources have shown that yes, animals experience emotions just as humans do. The studies may not be as frequent or as simple as human emotion studies, however evidence shows that emotions are no longer unique to humans, and are actually shown in animals as well.