In planetary and astronomical science, what exactly is, or is not, a tidal force?
The "tidal force" is simply a manifestation of good old inverse-square-law gravity.  It doesn't need a rigid body to be present to manifest.

Consider a giant circular ring of dozens of space stations established in interstellar space. They're far away from any star and just sit there happily doing whatever it was they were set up to do.  Suddenly, a star sweeps in from stage left (perhaps a fast-moving star created by a supernova-disrupted close binary system).  What happens to the ring of stations?

They are attracted towards the rogue star, of course and start moving noticeably towards it when it gets close enough.  They accelerate towards it pulled by the force of the star's gravity, which follows the inverse square law. But they don't all accelerate at the same rate. The station on the edge of the circle closest to the star accelerates fastest, stations half-way around the circle accelerate, but not as much, and the station on the far side of the circle accelerates least.

Soon the circle becomes an oval with its long axis pointing towards the star.  This is the tidal effect.  Let's say that the rogue star was dark -- a neutron star, perhaps, which had been stripped of its accretion disk by the explosion.  What would the people on the stations observe? They wouldn't notice the acceleration per se because they're in free fall.  Unless they were doing doppler spectroscopy of distant stars, they wouldn't detect their velocity (at least not at first), but they would see their neat ring of stations turning into an oval. From their point of view, some force was tugging on them and dragging them out of alignment. That's the tidal force.

From the point of view of the omniscient observer (us!) there are no tides, there's just simple gravity.  But from the point of vie of the freely falling observers in the gravity field, most of the effects of gravity disappears and what's left looks like a force.

Wikipedia has an article on the tidal force which gives a mathematical treatment. (Personally, I don't think the article is terribly clear, though it appears to be correct.)