The poem contains two main kinds of repetition.
First, it repeats a number of words that are central to its story and themes, including "beautiful," "lovely"/"love," "married," and "ring," plus the names of the main characters and the exclamatory "O." Clearly, this is a love story, and one with a happy ending! Then there's the repetitive phrase "hand in hand," which also relates to love—and is especially charming because neither owls nor cats have hands. (Paw in wing?)
Second, there's the structural repetition at the end of each stanza. Lines 7-11, 19-22, and 30-33 all repeat a phrase, partly or in full, four times over. Look at the end of the last stanza, for example:
They danced by the light of the moon,
The moon,
The moon,
They danced by the light of the moon.
These repetitions enhance the sing-song quality of the poem, making it more fun for kids to hear and recite. (Or even sing!) Here and in the first stanza, the song-like cadence is especially appropriate, because in each case, the characters themselves are singing or dancing.