How to pick up the car I customized in gta 5

I also bought it and the same happened to me. Although now after I've bought another property and they want me to do a mission for them a "P" pops up at their location. So I guess it's either bugged or it will appear later, my guess is on the first alternative.

(Edit) It seems like there is a third option; if you take too long they don't want your help anymore and send you another text "saying so". Check your phone for another text from your property.

Each character has a personal unique car. If lost, it will appear near the garage of his safehouse. Sometimes, it will be in the vicinity of the character, when you switch to him.

Any customizations made to it will be permanent and cannot be undone, unless you re-customize it.

At some point in the story...

Jimmy De Santa will "borrow" his father's personal car, at which point it will become unavailable until Jimmy brings it back.

Michael's car will be customized, so you might want to customize his car only after that event.

No matter what you do, you cannot permanently lose your car, and any and all customizations done to it is permanent. Do not be afraid to abuse your tuned personal car as a cannonball, and don't be afraid of losing it.
Basically, tuning your personal car is a safer investment than tuning any other car.