parshanut torah comment - Moses lived 120 years, and tradition says no one can live longer, so why do they?

Genesis 6:3: And the LORD said: 'My spirit shall not abide in man for ever, for that he also is flesh; therefore shall his days be a hundred and twenty years.'

In Deut. 34:7 the age of Moses upon his passing is given as 120, but most importantly it says "his eye had not dimmed, and his vigor had not diminished." To have one's mental and physical faculties—that is what we wish someone via "till 120."

These two verses seem to be the basis for the tradition that people should live until 120 but if that is the case why do people (especially in the Bible) live much longer than 120?

Answer

Although, as @DoubleAA noted, Ibn Ezra, as well as other commentators, identify the relevant verse (Bereshis 6:3) to be referring to the deluge and irrelevant to the human lifespan in general, still other commentators (Abarbanel and Malbim are ones I know of) interpret this verse as placing a cap on the human lifespan. But how do they reconcile this with the people (especially people in the Torah) who lived longer? I will quote Abarbanel on the topic:

ואין להקשות על זה מה שהקשה הראב״ע שאחר זה חיו בני אדם נח והאבות יותר מק״כ כי לא היתה הכונה בגזרה הזאת שמיד ולדורות יתחילו האנשים לחיות זה המספר מהשנים כי איך יתכן שיעתקו השנים מן הקצה אל הקצה פתאום אלא שילך חיות בני אדם ומספר ימיהם הולך ומתמעט בחסרון בירידה תמידית עד שיגיע אל הגבול ההוא ושם יעמוד הטבע האנושי ולא יוסיף

So not that the lifespan would revert immediately from the 800-900 range to 120, but rather that the lifespan would gradually decrease to like 400-500 years in the generations after Noach, to the 150-200 years in the times of the Avos, finally to 120 after the time of Moshe, who actually lived to 120.

Now what about the people after Moshe who lived longer that 120 years? (This means born after Moshe, for clearly Pinchas lived a very very long time, and this doesn't bother anyone.) Well, one could answer that Abarbanel and likeminded commentators weren't aware of anyone post-Moshe living past 120, and based their interpretations accordingly. But, I don't think this would even pose much of a problem to their interpretation. Another quote from Abarbanel (linked above):

ומשה בן מאה ועשרים שנה במותו לא כהתה עינו ולא נס לחה ועל זה נאמר לו הן קרבו ימיך למות כי היתה הקורבה ההיא הגזרה שנגזרה בכלל על בני אדם. וכן אהרן חי קכ״ג שנים שהם בכלל ק״כ שנה

Aharon, who lived to 123, lived "within the bound of 120". In other words, the limit is not 120 as much as "120 give or take", or "in the 120s".

The people who we find living past 120 do not live much longer than 120, maybe a few years at best.