> We're just their pets
I am not sure it is ever established that we are anything. I think in the Culture novels the term "human" is used as a shorthand for "humanoid citizen of the Culture", not necessarily Homo Sapiens (or their descendants).
I understand where that "pet" metaphor is coming from, but I also think it is very limited. It is looking at a relationship from the point of view of power level and intellect, and nothing else. The relationship depicted in the books has many more dimensions that are left out, and that I consider critical. I would highlight 3 items.
The first one is that even if Minds are (probably) as intelligent compared to us as we are compared to housecats, the fact that the "base intelligence" that we are using is that of a human already provides a richness to the relationship that the "cat" can not simply provide. A human could (for example) attempt to hide information from the Mind in order to further their own interests, months from now. Probably foolish, but a kind of foolishness which is outside of what a cat's planning capabilities.
The second one is on the opposite side: the relationship that a regular human can have with a housecat is limited also by the "highest-level mind". A Mind would be able to have way richer and more varied relationships than a regular human, including with other humans, merely because it has more options at its disposal. A Mind could (for example), track thousands of external body signs in order to deduce that a human is hiding something, and still have plenty of other options at their disposal (down to actual mind-reading; although that would be probably distasteful).
Finally, I think Minds need humans. This is not explicitly mentioned in the novels, but all the minds that had no humans with them (that I remember, at least) where the "unstable", eccentric or downright crazy.
I think a better relationship analog with something that exist today instead of a "owner vs pet" is a "parent vs child". The power balance is still there, the difference of raw mind power is also still there, but the parent is strongly motivated to make the best possible world for their child, and they need each other. It's just that in this case, humanity is the parent, and the Minds are the descendants. But well, I said "better", not perfect.