As mentioned in the following, the 'readings' of numbers are sometimes changed so that the readings mean something else. You can find more common mnemonic reading in here.
In the case of 2/22, it is simply reading 2 as にゃん, which starts with に. So, 2/22 is the longest possible consecutive 2's as a date, read にゃんにゃんにゃん which sounds like cats' meowing, hence Cat Day.
I just found that in 2022, there were even more 2's and it was called スーパー猫の日.
According to here
11/1 is 犬の日 - reading 1 as わん is common (the English number 'one'). The pun is the same. Practically this is less recognized (less advertised marketing-wise).
11/22 is ペットたちに感謝する日 by わんわんにゃーにゃー. Also this date is commonly known as いい夫婦{ふうふ}の日 ('Good Couples Day', reading 1 as い and 2 as ふ(う)).
(BTW according to the same webpage, 8/8 is World Cat Day).