He invented ‘undress’? Like he invented ‘undo’ or ‘unwell’? Come on, that’s silly.

Invented might be a bit strong, but he is certainly the first written record of the word. Dress existed as a verb already, as did the generic reversing “un”, but before Shakespeare there is no evidence that they were used this way. Prior to that other words/phrases, which probably still exist in use today, were used instead. Perhaps “disrobe” though the OED lists the first reference to that as only a decade before Taming Of The Shrew (the first written use of undress) was published, so there are presumably other options that were in common use before both.

It is definitely valid to say he popularised the use of the word, which may have been being used informally in small pockets for some time before.