It's not entirely ungrammatical to say I gotta make a plane meaning "to arrive in time for boarding" but since you ask whether people say it, the answer would be no, not in great numbers. If you put everyone who says it on a plane, there would be empty seats.
I assume such questions are asked not out of academic curiosity but with the desire to know whether the phrase would sound perfectly natural to a native speaker.
The usual phrase is:
I've got a plane to catch.
Change "plane" to "flight" and things are different. Then "make" sounds natural.
Can't stay, sorry. I've got to make a flight.
P.S. Subtle things like the determiner can have a significant effect as well. I would consider these idiomatic:
If you don't hurry up, you won't make your train.
If you don't hurry up, you won't make your plane.
But "make your flight" would be far more likely.