It's basically just a pop culture/fantasy trope. With that said, I know someone is going to come along and nitpick, so here's my hedging. Yes, it's true that various groups that could be considered "barbarians" have used axes at one time or another. The examples I'm most familiar with are the various Viking Age axeheads catalogued by Jan Petersen. But that doesn't make them a barbarian weapon!

Axes were in no sense dominant among "barbarians" to the exclusion of other weapon types. The various Persian groups that Rome fought with preferred bow, lance, and mace. The Roman-era Celtic elite preferred swords, while their rank and file mostly used spears. This remains true for the Migration Era Germanic confederations. Even among the Vikings, spears and swords were very common. In the late Viking Age, a particular kind of axe, often referred to as the Dane or Danish axe, become a renowned prestige weapon, not only among the Norse but among the English and northern French as well.

Tl;Dr: axes are useful weapons that have come in and out of vogue in different times and places. There is no reason to associate them with barbarism, or swords with civilization.

As you may surmise, this depictions are pretty much completely inaccurate, and as u/Rittermeister summed it above, it is primarily a fantasy trope, propagated quite recently, along with 'horned helmets'. I'm not an expert on 20th century fantasy art, but I suspect fantasy book cover artists from 70's and 80's to be the main culprits (along with J.R.R. Tolkien and his depiction of Dwarves visibly influenced by Norse sagas), as heroes (most notably Wotan and Siegfried) wearing costumes and props designed by Carl Emil Doepler for the 1876 premiere of Wagner's 'Nibelungenlied' (commonly, but not completely correctly considered the origin of 'horned helmet' trope) are almost universally armed with swords and spears.

A closes association between 'barbarians' and axes can be found in the Byzantine sources, where e.g. Anna Komnene in 'Alexiad' describes members of Varangian Guard as ' πελεκυφόρους βαρβάρους' or 'axe carrying barbarians', although the term 'barbarian' had a meaning closer to modern 'foreigner' rather than an implication of cultural inferiority. Furthermore, Anna also states that the Varangians were armed with variety of weapons, including swords and spears (cf. 'Alexiad', vol. 9, ch. 9-2).

Axe was, and still is a versatile tool and thus was ubiquitous among soldiers until recently, as wood was used to fuel fires, build makeshift defences, raise tents, repair weapons etc. This is why it makes sense to imagine marauding parties using this tools instead of other weapons, but historical and archeological evidence does not corroborate this notion. The trope associating swords with Romans and axes with 'barbarians' is completely false, as iconic gladius was derived from Celto-Iberian falcata after Romans met Iberians during Punic wars. Later Roman spatha, more useful in horseback fighting was derived directly from Celtic swords, so if anything, sword (read: cruciform slashing and stabbing weapon) was actually more typical for the 'barbarians' rather than Romans who focused on javelins, spears and shortswords made for stabbing from behind a large shield. Archeological findings show that Celtic swords were already popular in the Hallstatt culture period (8th century BC) and were very common among various Germanic peoples in the late Roman period and also vastly popular among Norsemen in early medieval period. Swords, rather than axes are being found in pre-Christian Germanic and Norse burial sites.

Contrary to what fantasy often conveys, one of the most common weapons in the field was a polearm that was used by virtually anyone. It was cheap, easy to produce, very useful for defence on foot and for mounted assault. Swords and similar weapons were more common as a side weapons, used in close quarters, when longer weapons became unwieldy. Axe, being a popular and ubiquitous tool, was also very common on the medieval battlefields among the retainers, who were common people obliged to accompany their master in the battlefield, and were often using tools they were accustomed to. English sources also note that English archers were often resorting to common tools in close combat, as most of them were equipped with axes, mallets and hammers to build simple defences. And given that 'knights' (read: heavy cavalry) formed only a small part of a typical medieval European army, you could say that axes were at least popular as swords, if not more. Both types of weapons are depicted as being used by both Normans and Anglo-Saxons on the Bayeaux tapestry, and multiple evidence shows that axes were also a common choice among mounted knights, either as main or secondary weapon. Notably, Richard I Lionheart was depicted as wielding a heavy axe in combat. Veracity of these accounts are disputed, yet they are a good evidence that axe was not considered a 'barbarian' weapon, as authors of these accounts saw nothing wrong with depicting it in hands of a king.