I think nursery rhymes are not part of modern culture at all. Kids probably never hear them and certainly aren't taught them. That one has several possible explanations, of which I choose to think the astronomical one is best. Constellations Leo (cat), Lyra (fiddle), Taurus (cow), Canis Minor (little dog), Ursa Major and Ursa Minor (the Big and Little Dippers dish and spoon) which appear close to the Moon.
Some nursery rhymes are a bit more sinister, like Rock-a-bye Baby, or falsely accused of racism (Baa, Baa, Black Sheep), or simply confusing like Hickory Dickory Dock. Most people think the latter is based on the clock in Exeter Cathedral, but some think it is about Tumbledown Dick (Richard Cromwell, son of Oliver). There is good reason (while I am
remembering my sheep) to see a connection in the rhyming structure to the way traditional shepherds count their sheep, using the Brythonic Celt numbers, where 8, 9 and 10 seem to fit awfully well:
Yan, tan, tethera
Methera, pip, sethera
Lethera, hovera, dovera, dick.
OK, While I'm on it, that version is the best known, from the north of England. Where I was born in Wiltshire, it was slightly different:
Yahn, tayn, tether
Mether, mumph, hither
Auver, dauver, dick
You only have to count to ten, then you transfer a pebble from one pocket of your smock-frock to the other and start again.