I think the reason this could reasonably be called 'private' is because the pilot is not going to be making any money out of it. The record of payments will quite clearly show whether the costs were shared equally.
I like the idea, but not as a 'taxi' type app, but as a way that qualified (or even people in the process of learning) can find another (fully qualified) pilot and pay to take a flight that a pilot may be going on anyway.
This would allow someone to get some experience of what happens on a variety of flights without having to fly themselves.
......by the way, a possible 'legal' test in these things is about control. i.e. the app cannot 'assign' pilots to passengers or vice versa and neither party has any obligations, the flight can be cancelled on a whim by either party with no penalty.