Synonyms aren't only words that mean the exact same thing. They may be words that mean approximately the same thing. Moreover, many do call their adult dogs their "puppy," just like many call other adults their "baby." Just as "baby" can refer to an adult person, so also can "puppy" refer to an adult dog.
Also, you should note that the entry doesn't cite those words as "synonyms." Rather, it says they come from a thesaurus. A thesaurus provides words of similar meaning whose closeness in exact meaning tend to decrease as one moves down a list. If thesauruses only provided words that meant EXACTLY the same thing, they wouldn't be very helpful as we often use them to try to home in on a better word, one that more exactly says what we mean to say.