This is a language question more than a history question, so I think I can answer for you. Avarice in English shares the same etymology as avar in Catalan, Romanian, and Occitan, avare in French, avaro in Italian, and something similar in the rest of the Romance languages. It comes from the Latin aveō, “I long for; I crave”.
If you are referring to the Avar people who inhabit Azerbaijan, Russia, and other places in that region, there is no connection between the two words — they simply are the same sounds in the same order, without any historical or linguistic connection. The name of the Avar people comes from their own language, belonging to the Northeast Caucasian language family — highly different from the Romance languages, and with minimal etymological overlap, if any at all.
Another example where the etymology of a word doesn’t come from the obvious source is the Dutch word for leopard, luipaard. The legend goes that someone discovered this big sleepy cat and said it was a lazy (lui in Dutch) horse (paard in Dutch). Well, no, actually — it was just imported from the French, léopard, which in turn came from Greek. So, even though there can sometimes be an obvious explanation, it isn’t always the right one, and especially when considering similarities across language, the answer to “why do these two languages share a similar or identical word?” is, with disappointing frequency, “just ‘cuz”.
Adding to the response of u/CPEBachIsDead, it is worth noting that the two words are extremely unlikely to be related on historical and lingustic grounds. Answering to the question about the relevance of the language, it is of crucial importance, because only then we can make any estimation of the possible lingustic transfer. The Catalan 'avar' or Corsican 'avaru' come from Latin 'avaritia' that is widely attested since at least 1st century BCE (the earliest I can name from the top of my head, as it was used by Horace or Lucretius). Now, the demonym 'Avar' is first attested only in mid-6th century BC, in the chronicle of Menandros Protector, Byzantine historian who described the arrival of 'Avar envoys' in 557, making it explicit that this particular nation was unknown to him and the court in general. The mention of their highly unusual braids worn by men also suggests that Avars were either unknown or barely known to Greeks at the time. With the second mention being written by the same chronicler in 576, this time mentioning the message of Turkic khagan Turksathos who asked for an assistance in capturing fleeing Avar subjects, suggesting the the arrival of Avars to Europe could have happened before the middle of the 6th century. This is corroborated by the 598 letter sent by kaghan Tardus to Emperor Mauritius, quoted in the later chronicle of Theophilactos Simocatta, where khagan says that 'during the reign of Emperor Justinian, small group of the young warriors from the Eastern peoples named 'Var' and 'Hunni' came to Europe and called themselves Avars [...] as among the Scythian peoples, Avars are known to be the most fierce'. It is also possible that the name might be a simple mistranslation, as Menander speaks of 'Warkhonites' and Theophilaktos of 'Var and Hunni', what might be possibly the same name.
In addition, though we render this word in English (and other languages) as 'Avars', the term was initially rendered by Greeks as something closer to 'Abars' (Aβαροι), what suggests the etymology derived from Mongolian 'abarga' i.e. 'snake', possibly a symbol or totem of this particular people or even the reference to the aforementioned braids. It is worth noting that in the Medieval Slavic languages, the name was rendered as closer to 'Ob'r', suggesting that it was initially closer to 'Abar' and only became 'Avar' due to the 'v<->b' transition.
Thus, the earliest mention of Avars in Europe is many centuries younger than the Latin word 'avaritia' itself, and the fact that this demonym could have been pronounced differently in the beginning, makes it very unlikely that there is any causal link between these two words.