Why does the Fast & Furious franchise have a chronological order different from the release order?

As you already know that the Chronology of Fast & Furious franchise is different from the release order but the reason for it is more complex than the chronology.

• Tokyo Drift has nothing to do with the first two films.
• Diesel was hoping to make more films in the Riddick franchise of sci-fi action pictures. To do that, though, he needed to get the rights for the character from Universal, which also produced the Fast & Furious movies. He agreed to cameo at the end of Tokyo Drift — as a promise of his return to the franchise in a fourth film (which eventually happened) — but instead of payment, he asked for the Riddick rights. Universal gave them to him, and another Riddick movie came out in 2013.
• In the end of Tokyo Drift, when Sean has taken the title of Drift King, he is challenged by a new racer — who turns out to be Dominic, who says Han used to ride with his crew back in the day. (Diesel only make this cameo and its sequel for the deal with universal).
• Han's character from Tokyo Drift was proved too popular in the audience. And Dominic's conversation with Sean also left a few unanswered questions.

• Now universal's chance to boost the franchise is by using all this in his favor, they can't resurrect Han but they can make the movie out of the Dominick's and Sean's discussion.

• And then franchise makes 3 sequels to first two films and prequel to Tokyo Drift. And the last Furious 7 was the sequel to Fast & Furious 6 and Tokyo Drift both to close the non-linear chronology.

Now every production house wants to make more money and Universal obviously got success in doing so. And this non-linear chronology is not so uncommon too. Mid-equal, prequel, and parallel are a way to do this non-linear chronology possible. Example are Star Wars, Saw (where two sequels play in parallel), etc.

Sources: Vox.com