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Is there a term for when a phrase in a song, i.e., "shivers down my spine" as in Bohemian Rhapsody, after that phrase is sung, a tinkling or some kind of sound is made with an instrument, to mimic "shivers down my spine"?

We call this text painting (or word painting), and it's a clever compositional tactic that dates back centuries. In short, text painting occurs when a composer does something in the music that is a direct reference to something in the text.

Text painting can be very literal, like when the text uses words like "rising" or "higher" and the melody ascends. But it can also be much more abstract, like when ambiguous harmonies mimic the poet/songwriter's increasing confusion (for whatever reason).

Franz Schubert was especially famous for using text painting in his songs and song cycles, but examples abound from long before Schubert and even into today's film scores.

Somewhat related, but not specifically about sung lyrics, is mickey mousing.
https://en.wikipedia.org/wiki/Mickey_Mousing

Mickey Mousing is when a piece of music that accompanies a film attempts to mimic the action's rhythm and pitch, similar to the literary idea of onomatopoeia.