Ursa Major
Ursa Major commonly known as 'The Bear' appears in a lot of ancient myths and legends on constellations. The more modern 'Big Dipper' is found within Ursa Major but I'm focusing on The Bear itself. The main story related to Ursa Major derives from ancient Greek mythology as is where most of the 48 constellation names come from. The influence of both the Greek and Roman cultures may be plainly seen; the myths behind the constellations date back to ancient Greece, but we use their Latin names.
Greek/Roman legend
Ancient Greeks associated the constellation with the myth of Callisto, the beautiful nymph who had sworn a vow of chastity to the goddess Artemis. Zeus saw the nymph one day and fell in love. The two had a son and named him Arcas. Artemis had already banished Callisto when she had learned about the nymph’s pregnancy and a broken vow.
However, it was Zeus’ jealous wife Hera, who was not amused by her husband’s philandering, who would do even more damage. Angered by Zeus’ betrayal, she turned Callisto into a bear.
Callisto lived as a bear for the next 15 years, roaming the forest and always running and hiding from hunters. One day, her son Arcas was walking in the forest and the two came face to face. At the sight of the bear, Arcas quickly drew his spear, scared.
Seeing the scene from Olympus, Zeus intervened to prevent disaster. He sent a whirlwind that carried both Callisto and Arcas into the heavens, where he turned Arcas into the constellation Boötes, the Herdsman, and Callisto into Ursa Major. (In another version, Arcas becomes the constellation Ursa Minor.) This only further infuriated Hera and she persuaded her foster parents Oceanus and Tethys never to let the bear bathe in the northern waters. This, according to the legend, is why Ursa Major never sets below the horizon in mid-northern latitudes.
https://www.constellation-guide.com/constellation-list/ursa-major-constellation/
Viking legend
One of the most well-known constellations is The Big Dipper, the Plough or Charles’ wain, which in the Nordic countries are called ”Karlavagnen”, the chariot of Karl or man. The wagon is easily identified, but the origin of the name is a source for discussion. One theory is that it is named after Charlemagne (Karl in Scandinavia) and of medieval origin. But if we look at the older Norse name, we find “Karlsvagn” (Man’s Chariot) but also ”Kvennavagn” (Woman’s Chariot) as the name of Ursa Minor. This might indicate a connection between the constellations, and the fact that the names are much older, with a possibility that specific names have been used during different periods in history. This means that Thor might be the Man and Freya the woman, both travelling by chariots according to the myths. We also might have a connection with the Nerthus cult, described by Roman authors.
Other legends
On the banks of the Euphrates thousands of years ago it was so designated, and the Iroquois Indians of North America called this star group "Okouari," their name for "bear." The Algonquin Indians called the constellation "the Bear and the Hunters," and as they were evidently sensible of the incongruity of attributing a conspicuously long tail to an animal that had none, they consequently regarded the three stars in the tail of the Bear as three hunters pursuing the beast. The Finns called Ursa Major "Otawa," a title resembling the "Okouari" of the Iroquois, and it is inferred that they regarded this constellation as representing a Bear. Thus in remote parts of the earth, in the far north, from the valley of the Euphrates to the region of the Great Lakes of North America, we find the same stars likened to an identical animal, "the relic of some primeval associa- tion of ideas long since extinct." The arrangement of the stars in Ursa Major in no way suggests a bear, or any other animal, and even if one nation should so picture it, there is no reason why the same imaginative creation should be universally identical. Aristotle held that the name was derived from the fact that of all known animals the bear was thought to be the only one that dared to venture into the frozen regions of the north and tempt the solitude and cold.
https://archive.org/stream/starloreofallage00olco/starloreofallage00olco_djvu.txt
Native American
Revolving Male (called Ná hookos Bika’ii by the Navajo — the Big Dipper)
Revolving Woman (called Ná hookos bi á adii — Cassiopeia)
Also referred to as First Man and First Woman, these two constellations are located on either side of the North Star or home fire. They are the official leaders of all other constellations. They never leave this area of the sky and no other constellation interferes with their routine. This arrangement of constellations established a law that has persisted to this day. This law stipulates that only one couple may live by one home fire.
https://starlab.com/wp-content/uploads/2017/04/D.-9.-Native-American-Myth-v616.pdf
The Iroquois Legend of the Three Braves Long ago, the Great Bear wandered freely throughout the sky. His massive paws took him far across the unlimited ceiling of the world. He hunted and fished, finding food there in the many rivers of the sky. All throughout the first spring he did this, until his belly was full and he felt happy. He did not know that three young braves had discovered him feeding that spring. They sought his pelt and meat to feed their families in the long winter that they knew was coming soon. Without warning, the braves ran out after the bear, trying to catch and kill him. The Great Bear ran, trying to escape from the hunters. All through the long summer he ran, always trying to get away. The braves, however, were very cunning and strong. Eventually they caught up with him. In the first autumn, their arrows pierced the Great Bear and he died. The blood of the bear spilled out of the sky and tinged all of the leaves with red and orange. The trees then dropped all of their leaves in mourning for their friend, the Great Bear. The Great Bear was reborn the following spring, as is the way of bears, and the braves set out after him again. They do this each year. If you look into the sky and watch, you can see the three braves trailing behind the Great Bear as he runs toward the horizon, only to do it again and again with the coming of each spring.
https://nieonline.com/downloads/national_wildlife/13wlw/bears_in_the_sky.pdf
Conclusion
It appears that the referring of Ursa Major to a Bear is due to the navigational use and positioning of the constellation. How it is useful to navigate by because it was always visible in the sky. This aspect of the constellation always being visible might also be referred to in the myths and stories as to punishment, not being allowed to touch foot on land or "bathe in the waters of the earth". This would make more sense as the constellation Bootes does drop below the horizon. There could be a punishment connection between the circumpolar constellations and Greek myth's. The Algonquin Indians story could lead to further interesting facts about the human reasoning for why elements about the planet happen like seasons changing and what that might impose. Also, the Native American story seems to help impose a moral law upon the people of that culture.
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Maybe other constellations and other cultures use these stories not just for helping to learn but helping to pass on morals to children and continue cultural teachings.