Akinator, the app and website, both display images of what I assume is copyrighted material. How do they manage to get around copyright law? Is this considered fair use?
There are several things going on with Akinator and their use of trademarked names and characters.
If you look at akinator.com Legal Notices it says:
2 – Copyright – Reproduction rights All the content of this site falls under French and international legislation on copyright and intellectual property. All rights reserved. Reproduction in whole or in part of this website, in any form or by any means is strictly prohibited without prior authorisation of the publications director. All products and trademarks mentioned are property of their respective owners. (emphasis mine)
The fact that they mention that all trademarks belong to their respective owners is good, but it's a formality.
What it appears the site really relies on for use of trademarked characters is what's called Nominative Use, which is a type of "trademark fair use."
The product or service cannot be readily identified without using the trademark (e.g. trademark is descriptive of a person, place, or product attribute). https://en.wikipedia.org/wiki/Nominative_use
Akinator has to use the names and trademarks to identify those names and trademarks. Akinator could possibly be sued to establish their type of fair use is not valid in their case, but it's probably not worth it for the companies to pursue. And, most companies would rather people use and see their trademarks in a marketing and money-making fair use sense rather than sue to stop their use.
See In the US, when is fair use a defense to copyright infringement? for a full description of Fair Use.
Now, when you get to the "answer" part of the game, you will see a "copyright" link. Part of that resulting page says
Displayed pictures are accessible from the web. It is Elokence's policy to respond to notices of alleged copyright infringement that comply with applicable international intellectual property law and make the necessary changes. In respect of Elokence's Intellectual Property Policy, Elokence operates a complaints procedure accessible via the form below.
So the company has given itself a further bit of protection by only using images on the web they can link to (at least in the web browser game; I don't know about the App), so that means the image is already "out there", uploaded by someone else, and Akinator can claim they didn't upload it. And, further, there is a contact form for trademark holders to send DMCA takedown notices for their work.
It's a belt and suspenders approach to the use of trademarks in their game.