Here is a video: Pay attention to "CCarCtrl::ListOfInterestingVehicles[0]" and "CCarCtrl::ListOfInterestingVehicles[1]" in Cheat Engine window. As you can see, taxi is removed from the list if taxi driver, who hasn't despawned yet, tries to enter it back, but driver door is blocked. This means that it turns into usual traffic car which despawns if player gets too far away from it.

Therefore to avoid despawn:

1. you can steal taxi in Luigi's girl, as you will safely reenter it in Don't Spank Ma Bitch Up
2. you can drive safely making sure you don't have left door blocked while driver still hasn't despawned
3. you can reenter taxi near Stallion to add it back to player's vehicles list.

One question: (assuming you get the taxi during dont spank) As far as I can tell, trying to get rid of the driver side door while the taxi driver still have not despawned (early during dont spank, on the convenient lamppost, for example) makes it so that the car is removed from memory and you need to go back into it at the docks.

At what point can we safely get rid of the door? Maybe on the road to the docks on the concrete ledge? Does the guy despawn by that time?

EDIT: According to my testing, you do not have to get back in the taxi if you pick it up second during Don't spank. You just have to not block the door of the taxi while you are getting away from the driver and it stays.

EDIT 2: Proof of concept:

vid of the earlygame, with a taxi that i get as a second car. i did not block the door, and it stayed without the need of getting in it again.

make sure to NOT block the door/lose the door close to the place you stole the taxi from. that is gonna make it get removed from the memory. see this vid:

Kind of extensive explanation of this mechanic (with commentary), including how to do it in runs:

First part of this vod is when the main stuff is said.