It sounds like FM synthesis[0] to me (so either an embedded FM software synthesizer or simply replaying samples of FM synthesis), so the timbre is quite unlike a "real" piano or PCM samples of a real piano. Timbre is a kind of "color" of "quality" of a sound - in my head it's similar to how different materials feel different to the touch. A violin and a trumpet can play the same "note" but their sounds have very different timbres. An apple and a ball might both appear red but have different surface textures.

The "piano" in the app sounds like the piano patches (presets) on FM-based synthesizers from the 1980s[1], where the designers tried to approximate the timbre of a piano using the limited (but fun!) possibilities of FM, while the transients and harmonics produced by a real acoustic piano are more complex.

The pitches and temperament sound OK to me, so I think what you are are hearing is the timbre. That is, the app is playing the "right" notes, but it's as if you're hearing a "different instrument", albeit one that is an early digital approximation of a piano sound.

[0] https://en.wikipedia.org/wiki/Frequency_modulation_synthesis

[1] e.g. the Yamaha DX7; here's one of its "piano" sounds for comparison: https://www.youtube.com/watch?v=F3rrjQtQe5A&t=214s

I have no problems explaining ie fb (f flat) or e# (e sharp) etc, in certain circumstances, but I'm a great believer in: don't mix the two (sharps and flats).