The ancient Greeks were familiar with the hot arid lands of Arabia and North Africa, the warm and relatively moist regimes of southern Europe, and the colder northern regions, so it was natural for them to place emphasis on the inclination of the sun's rays - in other words, on latitude - in differentiating between the various climates of their known world. Indeed the word "climate" itself comes from the Greek word klima which means "slope" or "incline".
Ceteris paribus, or other things being equal, temperature would indeed depend only on distance from the equator and the time of year. But ceteris, as life has always taught us, are never simply paribus; other factors ensure a wide variability of temperature even along the same line of latitude. In summer, for example, continental temperatures are higher than those over the ocean, and in winter the reverse is the case. These anomalies result largely from the differing thermal properties of water and of land.
In the first place, the albedo of a continental landmass - the fraction of incoming solar radiation that is reflected back to space - is different from that of oceans. And the albedos of various types of land surface differ from each other: soils and vegetation absorb most of the incoming solar energy, while snow and ice reflect it.
Another factor is that radiation falling on land is absorbed in a thin layer of soil and rock, and all the energy goes to heat the first metre or two below the surface. The energy concentrated in such a small volume of material produces a significant increase in temperature, while in the case of the oceans, vertical mixing of the liquid causes heat to be transported to much greater depths, and it is therefore diluted over a much greater volume.
There is also the fact that soil or rock heats up rapidly when exposed to sunshine, and cools down very quickly in its absence. Water, on the other hand, has a high heat capacity, and it takes a great deal of energy to warm even a small quantity of it by any significant amount. The temperature of the ocean therefore varies relatively little from summer to winter - in contrast to the continents whose temperatures change a great deal from the warm to the cold season.
And another factor affecting local temperature is the height above sea level, which even by itself is sufficient to cause wide variations between places only a short horizontal distance apart.
So the Greek concept of klima was very much an over-simplification, even if it was a good first approximation for its time.