The point is that when "alle" goes with a personal pronoun (like "wir alle" = "we all") that is in first position as a subject, it can be moved after the verb in informal speech, so: "wir alle lachen" → (inf.) "wir lachen alle" ("we all laugh"). In this context, though, I doubt that anyone would actually use "alle" in that position to refer to "wir", exactly because of the ambiguity that would follow, so I think it is safe to say that the meaning of the sentence is "all cats". Still, grammatically it could potentially be "we all", although "wir alle mögen Katzen" would be the first (and for most people only) choice of words for conveying "we all like cats".
In short: declension. "Alle" is a "der-Wort", which means it is one of a number of determiners and quantifiers that follows the same declension as the definite article der, die, das and also triggers weak declension in following adjectives.
"Alles" is the singular nominative & accusative neuter, it can be used with uncountable neuter nouns, but is most commonly found as a pronoun meaning "everything".
"Allen" is either dative plural ("wir geben Geschenke *allen Kindern" = "we give presents *to all children") or masculine singular accusative (to be used with uncountable masculine nouns).
"Alle" is either nominative & accusative plural or nominative singular feminine. It is also used as a pronoun meaning "everybody".
The singular forms are very rarely used because singular nouns are often accompanied by the definite article, in which case the form "all + der, die, das" is used (equivalent to English "all the" and used in plural too).
They have a very different meaning, the ambiguity in German stems from the flexible positioning of "alle": when it refers to a personal pronoun, "alle" can be moved after the verb in informal speech, which means that "wir alle mögen" (the standard phrasing for "we all like") can become "wir mögen alle". In many cases, "we all" is the only possible meaning, for example in "wir mögen alle Kaffee" "alle" cannot refer to "Kaffee" because the latter is masculine and singular. In many other cases, though, ambiguity can arise from shifting "alle" to third position, so this phrasing is generally avoided, which means that while in theory "wir mögen alle Katzen" can also mean "wir alle mögen Katzen" ("we all like cats"), one generally wouldn't say it like this precisely to avoid the confusion.
In conclusion, grammar allows "wir mögen alle Katzen" to mean "we all like cats", and if context was clear enough one might even choose this wording, but otherwise the sentence normally means "we like all cats".
Why does " the" need here in English? ; We like all of "the" cats.
"Alles" is singular nominative neuter and it it generally used as a pronoun meaning "everything", but it can also be used with uncountable neuter nouns. "Katzen" is plural (and it has to be, if you want to say "all cats") so the plural (accusative in this case) form "alle" is required.
Roughly, "all" looks at all the objects in a group together, while "each" or "every" focusses on all the objects individually.
"All the balls are blue" -- treats them as a group; uses a plural verb.
"Every ball is blue" -- treats each of the balls individually; uses a singular verb.
The meaning is otherwise similar.