I believe it was world building, you know? To help build upon the setting.

"The song is a plea by the crab Sebastian imploring Ariel to remain sea-bound, and resist her desire to become a human in order to spend her life with Prince Eric, with whom she has fallen in love. Sebastian warns of the struggles of human life while at the same time expounding the benefits of a care-free life underwater. However, his plea falls on deaf ears, for Ariel leaves before the end of the song."
~Excerpt from Wikipedia.
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While the song is a fun one, it did have a purpose, it was Sebastian's plea to Ariel, but she didn't listen, which is why it didn't achieve anything

I think it depends on how you define "achieving" something. A lot of Disney songs don't actually really cause any story progression, or character development - but they do give us insight into the character singing, and their view on the world.
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"God Help the Outcasts" from Hunchback of Notre Dame doesn't really cause anything to happen. It's just Esmeralda sending a plea to a being she's not even sure is listening to her. The story doesn't change because of the song, her situation isn't meaningfully impacted. But it does show us who she is, her relationship with the world around her, her views on religion.
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"Part of your world" from the little mermaid similarity doesn't *achieve* anything - but it does tell us what Ariel wants more than anything, and gives us a bit of insight into why.
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(Cont)

just can't wait to be King" from the Lion King *does* end in a distraction for zazu, but at the end of the day the song wasn't actually necessary to achieve that. The song doesn't move the story forward in any otherwise meaningful way - and really only serves to expand on something already established in the plot: that Simba wants to be king.
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But it does give a lot of insight into why he wants to be king (reasons that are largely childish, which ends up being part of his character development later on) - and it also is another opportunity to elaborate on the dynamic between him and Zazu.
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(Cont)

Under the Sea is much the same.
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Nothing *really* happens because of the song. But it does help express Sebastian's outright mistrust for everything above the surface - as well as his love for the ocean.
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In a way, serves as a counterbalance to Ariel's song "part of this world," which features the many reasons she wants to *leave.* - under the Sea, meanwhile is meant to highlight everything she is leaving behind. The music, the fish, the company, the clean water, the swimming, etc etc etc.
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And, as with "I just can't wait to be King," the song also expands on Sebastian's relationship *with* Ariel - and his devotion to her safety and her family.
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So, as a story progression song, it's not got a lot going on. But in terms of character building I'd say it has it's place. Definitely does a better job than a song like "shiny" (Moana) at any rate :P

What I mean is Ariel didn’t listen so he might as well not have sung it

She was a teenager and he was an adult trying to guide her on the path he thought was best in life. Of course she didn't listen.

Another thought;
In beauty and the beast (the 90’s animated picture) DaFoe is stuck in a snowman- this is foreshadowing of Josh Gad, who plays Olaf, playing him in the new live action version