This question seems to be asked again and again and always with the same basic assumption, i.e. that AI/robotics has become hyper-advanced and yet humanity has remained the same.
Imagine that you traveled back in time 2000 years and were trying to explain 21st-century politics and economics to the people back then. Imagine trying to explain the constitutional republic, the American tax code, derivatives markets, quants, market makers, and automated trading algorithms. Imagine trying to explain the concept of incorporation; or how products make money from advertising or freemium models. Imagine telling them that people make money from making videos and letting other people watch them for free. Do you think that it would make any sense to them?
I guarantee you that it's going to be the same within a century, let alone a millennium. We can't really imagine it, because the ideas haven't even been conceived of yet, or maybe they have but they're currently impossible to implement and therefore dismissed as crackpot ramblings. And that's just the economic system and monetization as a whole; there are obviously entire categories of professions today that were simply unimaginable hundreds of years ago (airplane pilot? food scientist? hand model? I'm intentionally ignoring anything computer-specific).
When it comes to our prospective relationship with superintelligent AI, it's basically impossible to predict because (a) there are so many competing theories, and (b) in a relative sense, we are total morons. We cannot truly comprehend the concept of a superintelligence any more than we can truly visualize a 4-D object.
But we want answers anyway. So at this point it's a matter of picking your favorite AI researcher and/or sci-fi author and running with it. Here's a sort-of canonical list of the various scenarios and how people think they might play out, in no particular order:
We messed up. We created defective or Un-Friendly ASI, or a large interconnected network of AGIs with Un-Friendliness as an emergent property, which is basically the same thing. We become a transitional species, a metaphorical biological bootloader for silicon-based life, fulfilling the same role as all the other hominid species before us.
It's not hard to imagine why AIs would be put to work on creating convincing simulations; just look at the enduring popularity of MMORPGs. It's also far more economical to live in a virtual mansion than a real one. A superintelligent AI might force us à la The Matrix, but I like to think we'd go willingly. In this scenario, we don't really interact directly with AI, we live in our own little humans-only world while the AIs solve the really hard problems like surviving a global cataclysm or Big Crunch.
We improve ourselves, with technology picking up where evolution left off. Genetic engineering (Iain M. Banks), external memory (Hannu Rajaniemi), multiple personalities as parallel processing cores or ultra-low-latency hive minds operating as a single consciousness (Peter Watts), or good old cybernetic implants (every sci-fi ever). There are many choices, but the bottom line is that robots never really outpace humans because we become advanced enough to improve ourselves at roughly the same rate.
In other words, the Borg, with or without the violent tendencies. It may still turn out that the best way to create artificial intelligence is to merge human biology with robotic components. Or maybe we see this as a route to immortality, biohack ourselves to such a degree that it's impossible to tell the difference between human and AI. The question of how we integrate obviously becomes moot at that point, as there's no "other" to interact with.
This is a sort of best-case scenario proposed by AI researchers, involving an AI/nanotech hybrid; a superintelligence decides that the best way to fulfill its Friendliness goal is to literally become the matter that we interact with - or a kind of invisible layer around it. This presumably would force major (probably positive) changes on political and economic structures, but in this case the AI isn't really a participant in the economy, and interactions with it would be sort of like interactions of the Enterprise's crew with the ship's computer. Humans still run their own economies, and the AI is just a facilitator.
Those are, broadly speaking, the most widely-speculated strong AI/superintelligent AI scenarios. You'll notice that not a lot of them incorporate the idea of "basic humans" interacting directly with human-level AGI, and that's actually because it would likely be short-lived if it happened at all; the current consensus (although it's far from unanimous) is a very fast take-off from AGI to ASI.
Humans just aren't very likely to end up living out their days as meat sacks, getting their meals from robot chefs, hitching rides with robot limousine drivers, and working for their robot
overlords bosses. That's the Hollywood version of AI, not reality. In reality, when AI technology reaches the level you're referring to, AI minds will be strange, alien, and largely incomprehensible to us. They might be capable of writing better symphonies or painting better murals than humans, but there's no reason why they would, just like there's no reason for me (an engineer) to data entry or tech support work. They'll be busy solving much bigger problems, and they're more likely to be distributed networks than cute human-shaped robots.
I get that you've probably already decided that you want to write about humans and advanced robots romping together, so not all of these may be helpful. That's fine, but if you're trying to construct a believable scenario, then you'll have to contend with the fact that it's not much more believable than humans romping together with dinosaurs. That doesn't mean you can't write that story anyway, but a contrived situation generally leads to contrived explanations, so pretty much just make up whatever excuse you want for robots not doing human jobs. (Personally, like I said, I'm partial to the "waste of their valuable time" explanation.)