Yes, huge liberties were taken with this phrase. The English phrase roughly corresponds to: "Can you think up a group of caged animals who are dreaming of handling fictional caged animals?"
The Japanese one, however, is more or less the following:
斜め七十七度の並びで泣く泣く嘶くナナハン七台難なく並べて長眺め
Naname nanajyuunana-do no narabi de nakunaku inanaku nanahan nanadai nannaku narabete naganagame.
And Black Hanekawa's kitty version is something like
にゃにゃめにゃにゃじゅうにゃにゃどのにゃらびでにゃくにゃくいにゃにゃくにゃにゃはんにゃにゃだいにゃんにゃくにゃらべてにゃがにゃがめ
Nyanyame nyanyajyuunyanya-do no nyarabi de nyakunyaku inyanyaku nyanyahan nyanyadai nyanynaku nyarabete nyaganyagame.
Now, very roughly, this translates into English as something like...
I have been watching these seven 750cc motorcycles (nanahan) which are noisy like horses, and are placed beautifully at an angle of 77°.
It's very clear the English and Japanese tongue-twisters are nowhere close in meaning. What I can say about why they were chosen is that the English one plays off the sound mee as mew, the sound of a kitten meowing; the Japanese one plays off na as nya or nyan, a common way of making onomatopoeia for a cat's meow.