I'll preface by saying this is not an area I am well studied in, so I hope someone with more knowledge on the subject will correct me if I'm wrong on something. Anyway:

While the two peoples you speak of, the Huns and Mongols, most certainly did not speak the same language, and in all probability did not have the same exact cultural traditions as eachother, they probably lived in much the same way. The nomadic horse archer lifestyle is incredibly old. Like, seriously old. The Roman Empire was not the first to experience raids by nomadic horse peoples, nor was Attila the only one to lead his people into settled lands. The Assyrians of Mesopotamia, who had their heyday thousands of years before the Romans even enter the scene, famously fought with similar nomadic horsepeople called scithians if memory serves, who were very similar to Atilla's huns.

Europe actually experienced a second invasion in the 9th and 10th century (a long time after Atilla and roughly three hundred years before Genghis Khan) from nomadic steppe people called the Magyars, and these were indeed very similar to the other two groups. It is also worth remembering that any civilization living close to the eurasian steppe probably dealt with smaller groups of nomadic peoples much more frequently; the large-scale invasions of powerful leaders like Attila and Genghis Khan are merely the more famous ones. The Chinese dynasties spent much of their time and effort trying to keep the steppe tribes from unifying in order to prevent exactly what Genghis Khan did.

In short, throughout all of recorded history up to the modern period, nomadic horseriders have populated the steppe and been a smaller or larger nuisqnce to the settled societies that they happened to border. This lifestyle was pretty much the most effective way to survive on the steppe, and up until modern firearms, mounted archers were supremely deadly on the battlefield (outside of battle, the bow and horse combination was primarily used for hunting in order to survive, but it turned out to be pretty effective at hunting people as well as beasts). As a result, there was not much reason to change that way of life all that much over the ages. Because it worked perfectly fine as it had for thousands of years.

In conclusion: While the Huns and Mongols are nearly a thousand years apart and were by no means the same people, how they lived would indeed have been very similar if not almost the same.