Running Redis on the same server still adds overhead to the system, to store an object in a map that's on the Java heap doesn't require any inter-process communication or serialization. Plus it's another point of failure, what if the Redis process stops but the Java process is still running? If Redis is remote, what happens if the network goes down? Keeping everything within the JVM avoids having to think about those failure scenarios.
I don't know how Mailinator is coded, but I'd guess he hasn't written anything too custom for storing key/value pairs in memory, there are plenty of Java caching implementations, the key thing he has done is work out a way of reducing the duplication, which Redis won't provide out of the box.
Mailinator is a great example of having the minimal set of features required and no more. There's no logins, no setup steps and no guarantees. This allows Mailinator to run on such minimal hardware, adding disk persistence would add complication without making the product better, by definition the kind of emails you send to Mailinator aren't important, so if they're lost during the occasional restart it doesn't detract from the service.