Hakuna matata means "no..." Uh, that is, it means "here there are no concerns". The phrase is in Swahili, and, no, it was not made up for the movie. In fact, a literal Kiswahili translation of "no worries" would be "hamna shida". Hakuna matata is a rather common Swahili phrase, and has appeared in plenty of other places outside of the movie whose name I have narrowly avoided mentioning thus far.
You can take apart the phrase as follows:
haku: Locative prefix; "there are not".
na: Negative suffix.
ma: Plural prefix.
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