You're both right! You're just talking about slightly different things. On the one hand, the city-state (broadly defined) is something we can identify throughout recorded history in many parts of the world, from Meso-America to Italy to India. The best modern example is probably Singapore. On the other hand, in every period and region, there would be a slightly different meaning to the concept, and we can't speak about them all as a single category to which the same rules apply. Your friend is referring specifically to the Ancient Greek city-state, which is a thing unique to its time and space - so much so that many scholars of the period (including myself in my more recent AskHistorians answers) no longer refer to these states as "city-states". Hurray, confusion!

In the 1990s and early 2000s, a research school known as the Copenhagen Polis Centre dedicated a huge amout of scholarship to the question how to define a Greek polis - the word that is conventionally translated as "city-state". A lot of the work they did was comparative, looking at city-states in other places and periods to see if there were any unifying features to these entities and if anything could be said about what was typical for all of them. But the result was mostly negative. While it may seem obvious for us what "city-state" means - it's a city that's also a state! - it turns out that this concept is actually very difficult to describe precisely without including things that aren't, or excluding things that ought to be counted as, city-states. What if a city-state is too small and rural to have a major urban centre (like Sparta)? What if a state consists only of a city, with no hinterland (like the Vatican)? What if it is not entirely autonomous, but part of an empire (like the Phoenician cities under Persian rule)? Which are necessary requirements, and which are optional?

In the end, it seems city-states have different features in different periods (in terms of their size, economy, form of government, level of autonomy...). What makes a city-state is to some extent a matter of opinion. And it doesn't even always help to look at the way historical societies themselves talked about their states.

Take the Greek word polis. It's conventional wisdom that this word can be directly translated as "city-state". But that's not actually the case. Its roots are in a Proto-Indo-European word for "fortress" or "citadel" - no more than the fortified centre of a community. In early Greek texts like the Odyssey, it clearly still has this meaning. But it also means the urban centre around that citadel, and the combination of city and hinterland that encompasses the territory of a Greek state, and the community of citizens living within that territory. The exact meaning of the word depends on the context and, to some extent, on the person who is using it. They would privilege different aspects of the concept in order to get their point across. Thucydides, for instance, famously has one of his speakers claim that not land or ships or walls, but men make the polis - no physical entity, then, no "city-state", but only a community of people. This idea allowed the Athenians, for instance, to abandon their city and land to the Persians without losing their cohesion as a community. On the other hand, the travel author Pausanias scoffed at a small place calling itself a polis, because it lacked all the physical trappings that came with the title - a marketplace, fountain houses, theatres, temples and the like.

In short, the word actually has multiple overlapping meanings. Some of these don't require anything that we would call a "city-state". Indeed, some major (and many minor) Greek states don't fit any concept of the city-state that requires there to be an urban centre controlling a rural hinterland. The average Greek polis had fewer than 1000 citizens scattered about its patch of agricultural land. These communities wouldn't have either the resources or the need to build what we would recognise as a "city" from which to rule their "city-state". When we apply this term to them, we are evoking an image that might fit Athens, with its sprawling fortified city and harbour, but that would have little to do with how most Ancient Greeks actually lived. And even at Athens, the great majority of the population lived in the countryside, in small villages, sometimes days away from the urban nucleus where the instruments of government were.

For this reason, experts like Mogens Hansen (the former leader of the now-defunct CPC) have largely abandoned the use of the word "city-state" to describe Greek political communities. They prefer the term citizen-state, which emphasises the centrality of the human element that was essential to Greek self-definition of their communities. The citizen-state is one that is defined by its citizens living together as a community, determining their own laws and ways of life, maintaining their own cults and customs, and deriving their basic identity from their part in it. This notion is less relevant to city-states in other periods, where citizenship need not imply direct involvement in government, and where the existence of the community is more a physical than a conceptual reality. It is often compared to Medieval Swiss cantons - though these were not urbanised in any way, and are not thought of as city-states. For the Greeks, the ideological reality of their community was the main thing, and it allowed even tiny backwater communities with no urban centres to rightly call themselves poleis.

This is what I mean when I say you're both right. Yes, there are many things throughout history that we call city-states, and there are serious comparative histories trying to pin down the concept, what it is, and how it works. But also, the Greek state is a singular beast, which has seen no exact parallel in history, and defies easy comparison to other political units.

Here's a related question about city-states being an anachronism now. Is San Marino still a city-state, or is it anachronistic to call it that? If it isn't a city-state anymore, when did it stop being a city-state?