My spanish is not perfect...
"On the taxi" - algo que está en el exterior del taxi. Something on the exterior of the taxi.
"In the taxi" - algo que está en el interior del taxi. Something on the inside of the taxi. Or the state of being within the interior of a taxi.
"In the taxi" is a more common phrase.
"I rode in the taxi."
"On the taxi" is uncommon.
"I saw a red light on the taxi."
The most common phrases around taxis...
"I took a taxi."
"I need to take a taxi."
"I need to call (or get) a taxi."
"I will have to take a taxi."
etc.