It is possible to name a pet with a space in the name - making it impossible to change the name after. A solution would be to disallow/replace spaces.
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it's not impossible to change the name after, i've done it myself.
/name cat kitty kat
You changed the name of 'cat' to 'kitty kat'
/name kitty kat jemima
You changed the name of 'kitty kat' to 'jemima'
Didn't even need to use quotes.
/name jemima super duper cat
You changed the name of 'jemima' to 'super duper cat'
/name super duper cat harrietta henry
You changed the name of 'super duper cat' to 'harrietta henry'