Playing pairs of non-cat cards
The instructions that come with the game have the quote in the question and indicate that there are basic and advanced rules.

The basic rules state that cards without instructions must be saved and played in pairs. They say that "if you play matching pairs with no instructions on them," you can randomly steal another player's card.  With the later section, it is clear this only applies to "cat" cards with no instructions.

There is a later section "for advanced play" which specifically states that "play pairs .... no longer only applies to cards with no instructions on them, it now applies to ANY cards in the deck with the same icons in the corners".  It also gives the rules for three of a kind play and 5 different cards play.  This seems to be the only set of rules listed on the "official rules" webpage in the other answer.

Ergo: Whether you can play non-cat pairs is based on which rule set you are following.