The passing of the Mark is a big deal in the story. Koon was entrusted to have it by the previous Keeper. The Keeper was in charge of holding the balance of the universe, so it was important that Koon was chosen and was trained to be aloof for the job to maintain control.
At first they tried to stop Hiisa from having the Mark because he was part of the White Line, the power faction that passes down the Mark belongs to the Black Line. Simply put, Hiisa was not meant for the job of Keeper. Later on as more backstory was revealed, the Black Line considered Hiisa too unpredictable of a force and too obsessed with Koon. He also challenged the status quo by being popular with the White Line, so they knew he was dangerous. He's unpredictable and hard to read, to challenge Koon for the Mark, it meant he wanted power.
Albeit, as the audience we know that it was his way of "freeing" Koon, but in Amber, they saw it was his method to simply gain power, maybe as a way to shift power between the Lines too.
I mean I get it, the obsessive love. White hair dude grew up with his identity wrapped up always watching his enemy. They both wanted to do something selfish.
The jargon was confusing. And I still don't get why he didn't just give the mark up? Or was that supposed to show his feelings toward white haired dude?