How are data types related to information theory?
Interpretation: The Link Between Data Types and Information Theory
I believe the missing link between these two topics, information theory and data types, is interpretation. Interpretation is essentially the "input-output function" for which output (meaning) is derived from input (data). Whether the interpretation is being performed by a computer, at compile-time or run-time, or a human is relative to the task at hand. 

At any rate, data types are language-defined/user-defined interpretations of an input alphabet to an output alphabet. Also, language-defined data types usually have a direct relation to the size of storage locations on a computer.

Probabilistic vs. Deterministic "Mapping"
When determining the output based on a given input, is the output definite (deterministic) or is the output uncertain/ambiguous (probabilistic)?

Memory vs. Memoryless "Mapping"
When determining the output based on a given input, does the determining process require memory?