This does not mean “to be gentle to your machine”. It is not associated with the sympathy definition that’s to do with “feelings of pity and sorrow for someone else’s misfortune”.
Another meaning of sympathy is “understanding between people; common feeling”. And so, mechanical sympathy means “to understand your machine”.
Mechanical sympathy is when you know the slightly odd movement that eases it into 2nd without drama, the small tells that warn of impending traction loss before a slide comes on, the feeling that some component isn’t doing its job and may be leading to failure. It’s an understanding between you and your machine that can only be formed through time together.
Every time I see the phrase used, it's used incorrectly :)