No. It goes farther.

In Hercules, the Greek gods of old exist. In the Little Mermaid, Poseidon is the father of Triton, king of Atlantis, meaning the little mermaid occurs in the same universe as Hercules, but some 2 millennium later, which would mean it also is in the same universe as the Lion King.

Here's the thing - across all three movies the same basic rule set can be seen: animals can talk to one another. Humans can talk to one another. But animals can't talk to the humans - Flounder can't communicate with Eric, Hercules can't talk to Pegasus. Sentient mythical or magical beings, however, can apparently communicate with both, acting as a communication bridge between the two groups - Ursula and Ariel can talk to Flounder and Eric. Sadly, this is not PROVEN in Hercules, but if one counts Pegasus as a mythical animal, we can see that while Hercules and Pegasus can't interface with the same language, they can still get the gist of what's going on, which would make sense as Pegasus is something of a cross between a magical being and an animal. This relation, obviously, does not appear in the Lion King, but between Scar's appearance in Hercules and the trend seen across the three movies in human population terms (each latter and latter movie displays a larger and more advanced human civilization, from none to ancient culture to colonial culture), there is a (strained) relationship that can be traced.

Using this (admittedly QUITE strained) train of thought, we can see Mulan could feasibly occur in the same universe on the other side of the world - Mulan can't talk to the cricket or her horse (and is never shown as acknowledging them as sentient beings), but Mushu frequently communicates with the cricket and horse AND talks to them as if he can understand them. As a magical creature, he fits the criteria for this type of communication, and as a family spirit, his existence doesn't contradict the existence of Greek gods. Instead, its quite possible ancestor spirits exist in the world of Hercules, which isn't TOO far fetched as normal humans like Meg can defy (at least age-related) death, and can in fact come back from the death if fetched from the underworld. Perhaps in that universe, the ancestor prayers of the Chinese allow them to communicate with the family spirits of the living world from the underworld.

But back to human/magical beings/animal relations. Going down this path, we can see Tarzan, while omitting the magical creature communication connection, follows a similar train of thought. On a completely different side of the world (roughly around the time period of the little mermaid), a human can be raised as an ape and learn to communicate with humans AND animals. In a universe where, say, magical beings were slowly losing traction as human civilization expands, its quite possible humans could be raised in an animal society and thus be able to learn communicate with animals (which are sentient beings unable to communicate with humans, including a non-ape Tantor), then learn to understand and communicate with humans. Thus, if we include Tarzan (and i cannot stress how strained this and Mulan are) in this universe, we can see that the barrier between man and animal, while existent and often traversed by mythical beings, can be overcome.

Which brings us to our final link in the chain, where I must reference the Aladdin theory, where Aladdin takes place 8,900 years in the future due to the Genie commenting on Aladdin's third century clothing and referencing 20th century culture after saying he's been trapped for 10000 years. Others have detailed this quite well, but some general trivia is that the passage of time has corrupted the Islamic faith to the point where people shout "praise Allah" as a meaningless phrase, and the architecture is nothing but a bastardization of Arabian culture at its height. In this same universe, communication between man and animal has progressed farther then in had in the time of Tarzan and (to a lesser degree) Little Mermaid, as evidenced that Aladdin has no problem understanding Abu, and Abu being treated as a sentient being (the guards recognize he is dangerous with a sword, Jasmine isn't disturbed by Aladdin's human-esque treatment of Abu) due to the extremely long passage of time. This is also the same universe where the Greek gods once existed and family spirits are a thing, so the idea of a magic carpet and all-powerful genie of the lamp makes sense within these rules. In fact, the cave of wonders would simply be a collection of the treasures of the old world, only allowing magical items within (the carpet, the Genie) to be used by man, the punishment brought on man for the gradual replacement of magic and mysticism in that universe.with the spread of human advances.

TL;DR? I have way to much free time and The Lion King, Little Mermaid, and Hercules are linked.