Learn English – A word for “taking something/someone serious”

Would an idiom do? "Steal one's thunder" might interest you.

Steal one's thunder: to grab attention from another especially by anticipating an idea, plan, or presentation
also: to claim credit for another's idea.

[Merriam-Webster dictionary]

There is the saying many a true word is spoken in jest. As Cambridge Dictionaries explains, this is

said about humorous remarks that contain serious or true statements.

I've seen this used in two different kinds of situations: The first is similar to that in your quote, where someone is trying to cover up the truth of their (usually insulting) statement by pretending it's a joke. The second is when a truly humorous story or statement also contains some important truth, such as a comedic movie that helps illuminate some part of the human condition that might be too painful to approach in a more serious manner.

The origins of this exact phrasing are murky, but similar sentiments have been expressed by many authors, including Chaucer and Shakespeare (see linked article by The Phrase Finder). It's also sometimes shortened to there's truth in jest, perhaps modeled after the similar English translation of the Latin aphorism in vino veritas (in wine, there is truth/there's truth in wine).