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This is terminology question, I know what is hinge loss mathematically; but I can't grok what hinge mean in hinge loss. Is it rod, or door hinge, or something else?

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It’s a door hinge, as seen in the shape of the loss function:

enter image description here

From https://www.vocabulary.com/dictionary/hinge:

A hinge is a type of joint that attaches two things together while allowing for limited movement. A door hinge fastens the door to the wall and lets the door swing open.

https://i.sstatic.net/Ifeze.png

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