You are right, Ajax is not right wing, Feyenoord does have some right wing asswipes in their fans but they will never meet without hmindless violence.
The Ajax fans don't see Feyenoord as the "Enemy" but as a unworthy often competition, it is Feyenoord fans who see Ajax as the enemy. Ajax fans have more animosity for PSV Eindhoven. The violent nature of Feyenoord fans can be seen in their violent conflicts with other clubs. Especially German clubs and other foreign clubs. Especially if they have contacts with Dutch clubs Feyenoord hooligans hate.
Every year that Feyenoord plays international matches you always have fines for violent actions from Feyenoord fans.
Now I wrote above here that Ajax does not see Feyenoord as a big thread but we still like to win LOL.
In the past 30 years Feyenoord was champion of the Eredivisie 3 times (eredivisie is our premier league to so speak)
In the past 30 years Ajax was champion of the Eredivisie 12 times and they are on their way for number 13
Ajax was been champion 35 times (that is why they have three stars in their logo. As soon as they become champions 40 times they will get their fourth star above the logo.
Feyenoord has been champion 15 times (thus one star above their name)
When Ajax and Feyenoord play the most likely team to win is Ajax. Since the beginning of 2000 Ajax-Feyenoord or Feyenoord-Ajax had the following results
a draw was the result 13 times
a victory for Feyenoord was the result 7 times
a victory for Ajax was the result 32 times
As you can see, we (as in the royal we, as an Ajax fan) have little or nothing to fear from Feyenoord. In the past 52 matches Feyenoord has just won 7 and that angers the Feyenoord fans, who think Ajax is arrogant (which is a bit true). That is why the hatred for Ajax is so large in Rotterdam/Feyenoord.
It does not help that Ajax has the bigger stadium, Ajax has about 10.7 million followers on social media, Feyenoord has 1.7 million.
About 50% of all Dutch people are fans/support Ajax,
second on the list is PSV with 22%. Feyenoord is 4th on that list, far behind Ajax.
Ajax has a budget of 160 million Euro, Feyenoord has a budget of 65 million. The transfer value of all the players on the field for Ajax is almost 350 million euro's. The transfer value of Feyenoord is 134 million.
Feyenoord has 4.8 million in the bank but have a loan of 17.5 million which has to be paid in the next 3 years. Ajax has a bit more in reserve. They have 224 million in their bank account.
In other words, Ajax has more fans, is more successful, a bigger budget, much more net capital, a more valuable selection, more profitable, better stadium, etc. etc. etc. etc.
And that is the reason Feyenoord hates Ajax's guts (envy mostly) and also the reason why Ajax does not feel that hatred for Feyenoord.