This depend on what civilization.
Aztec or Maya : definitely not because they did not use and in popular culture they are without horses, and this work well in other Age of game.
Lakota : definitely yes because they use it and it was important part of their culture and in popular culture they are with horses.
Inca : they don’t use horse but instead of horse they can use : llama cavalry
Also we can find alternative to horse, in the old world we ride : donkey, horse, camel (both arabian and bactrian) and there are even buffalo race in Indonesia.
In the new world we can look at prehistoric animal because most of them lived alongside humans.
For example imagine a Mapuche civilization, they must have mounted unit with spear, bow and bolas. But we can replace the horse by a prehistoric animal who lived in South America, the macrauchenia and for represent them not using modern representation without trunk but older one with trunk.
Imagine a mapuche rider with bolas who charge but instead of a horse he ride this : File:Macrauchenia.jpg - Wikipedia
Also if they put a Celtic mythology the celts will be use horse but like the egyptians who use horse, camel and elephant they can use another animal in addition : Irish elk, also named Megaloceros (there are depiction on this on internet, search with the words : Megaloceros rider) This animal was depicted in lord of the ring ( Thranduil ride a megaloceros in the movie The battle of the five army : https://static.wikia.nocookie.net/lotr/images/8/8e/Thranduil_in_Elk.png/revision/latest?cb=20190711135508 ) and also their is a chariot pulled by a megaloceros in the game Paraworld : https://vignette.wikia.nocookie.net/paraworld/images/b/bf/Norseman_Chariot.png . (This unit was in a norse themed faction who have also mammoth and wooly rhino unit)
So a celtic cavalry can use a mix of horse and megaloceros.
For a slavic mythology they will also use horse because this civilization must have a bogatyr unit this heroes use horse (there are nice XIX th century painting of them on Wikipedia). But for this mythology we can also use animal tamed to the cold because Russia is a very cold country, part of their cavalry can be reindeer (maybe with sled) and mammoth ( the word mammoth come from Russia).
For a mesopotamian mythology initially they don’t have horse and use onager. So we can put a heavy charriot pulled by onager like this : File:Ur chariot.jpg - Wikimedia Commons
For them a mix between onager (heavy cavalry) and arabian camel (light cavalry).
For a subsaharian african civ they use horse in the Sahel and have nice war cavalry, we can use them but better : with zebra instead of horse and maybe a rhino cavalry.