I was imaging a system which has a black hole, a star and Earth like planet. The star and planet orbits around the black hole, but the planet has same orbital period as the star, which is a "quasi planet" of the star (see:this, at this case, the Earth is replaced by the star and the quasi satellite is replaced by the planet). Is this planet habitable for life?
The orbits of quasi-satellites don't stay that way in the long term (see here: https://en.wikipedia.org/wiki/Quasi-satellite).
"Over time they tend to evolve to other types of resonant motion, where they no longer remain in the planet's neighborhood, then possibly later move back to a quasi-satellite orbit, etc."
This means that the star-planet distance would change drastically over time. During quasi-satellite epochs the planet's climate might be relatively stable but then it could freeze over for a long while, then perhaps return to a stable state.
A more promising setup could be simply to have a planet orbit in the star's L4 or L5 Lagrange points. If the Sun's orbit around the black hole were 1 AU in size, then Earth sitting at L4 or L5 would be 1 AU away. I think this could work, although there would be really strong tidal effects from the BH that could potentially lead to stability issues. Maybe this would work better with a brighter star so that it could be farther away. E.g., if the star were 16 times brighter than the Sun then you could scale up all the orbital distances by a factor of 4.
Of course, the simplest would just be to have the planet orbiting the star orbiting the BH....
Quasi satellites can get very close and very far from its host, essentially acting like an eccentric orbit. However, I am not quite sure if they have a simple periodic movement. My guess would be a more erratic movement, thus unpredictable seasons. Additionally, this type of orbital mechanics would cause the spin of the planet affected. Not just these, there would be a lot strangeness in that planet.
For habitability, it is possible to have life on the planet, but it would probably be under the seas as the temperature difference on the surface would be too high. If you set the position of the planet right, it would have liquid water.