The terms nearside and offside predate cars. Horses have an offside and a nearside too. A horse is invariably mounted from the nearside or the horse’ left. The reason, I believe, dates back to men carrying swords which, for a right handed man, are worn on the left and the sword would get in the way if mounting from the right.
In practical terms horses get used to always being mounted from the nearside so might get jumpy if you tried to mount it from the right.
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