Toobworld Central has no problem in whichever Penguin look was used in this blipvert, because like the Batman, "The Penguin" is an alias. (In the comic books, his true name was Oswald Chesterfield Cobblepot - as a fan of name-coining, I always hated that name.)
Since these Bat-blipverts are a continuation of the TV series and have no connection to the movies (save in their style), this could play out along two different storylines.
1] The Penguin could be the televersion of Danny DeVito's interpretation in "Batman Returns"
2] He could be the son of the original Penguin.
There could be other options, of course, including one that even combines those two suggestions. After all, nothing says that just because we adopted the look of DeVito's Penguin, then we have to also accept the character's origin story. Abandoned by his parents and raised by penguins? "Don't be ridiculous!" as Balki would say.
When we first met the Penguin as played by Burgess Meredith (a member of the TV Crossover Hall of Fame, by the way) in the second two-part story of 'Batman', he was already in prison, awaiting release. With the crime for which he was convicted, the Avaricious Avian may have been exposed to harmful radiation from whatever nefarious trap he had set for the Dynamic Duo. This would have altered the genetic code stored in his loins which he would have then passed on to any descendents.
It's a theory of relateeveety, anyway......