Why do most serializers use a stream instead of a byte array?

It means you can stream to arbitrary destinations rather than just to memory.

If you want to write something to a file, why would you want to create a complete copy in memory first? In some cases that could cause you to use a lot of extra memory, possibly causing a failure.

If you want to create a byte array, just use a MemoryStream:

var memoryStream = new MemoryStream();
serializer.Write(foo, memoryStream); // Or whatever you're using
var bytes = memoryStream.ToArray();

So with an abstraction of “you use streams” you can easily work with memory – but if the abstraction is “you use a byte array” you are forced to work with memory even if you don’t want to.