Surely opening up a new API doesn't give them retroactive rights to the name in that space.
Technically, Kik (the company) registered their trademark in the class "Computer Software" . That means that no-one else can use the word Kik (and the logo) for this class of activity. The key issue is when the registration happened. Since they've been going since 2009, and Kik (the software project) only started in 2015, Kik the company was there first.
The reason their lawyers are asking for "Kik the project" to change name is because trademarks can become generic if you're not seen to protect your mark . As a private company (particularly VC funded), IP has a lot of financial value: the business will be valuing their trademark. If they don't protect it then they'll be "throwing money away".
The fact that the business world (and VC's generally) highly value IP is why most people over-register. If they didn't go for wide classes, and then in a few years decided to work in a particular area they might not have rights to their "own name" in that class.
It's a case of the Open Source and Business world-views clashing.
 Someone else in the thread found the specifics but the link doesn't work for me.
 Everyone knows the example of Spam.
Please show an example of someone other than Hormel Foods Corporation marketing a meat product using any derivative of the name 'Spam' and getting away with it.
You don't seem to be very well informed.
That is not what happened here though.