What's the term for expressions like "man's man" or "lawyer's lawyer"?
According to "Construction Grammar" (schema-based grammar), they are a type of constructional idiom [X's X]. Their concreteness level is almost abstract because they are not that lexically specified.
From the book "Functionalism and Formalism in Linguistics: General papers" edited by Mike Darnell:
Another explanation from the book "The Reality of Linguistic Rules" By Susan D. Lima, Roberta Corrigan, Gregory K. Iverson:
They can also be a representation of formal (schematic) idiom.
From the article "Language and Cognition II Construction Grammar" by Prof. Holger Diessel:
They fall under "genitive compounds" (descriptive genitives) also. So it can be called an "idiomatic genitive compound" as well.
Here is another passage along with the description of "constructional idiom": (from the book "The Grammar of Words: An Introduction to Linguistic Morphology By Geert Booij"
I'd call that type of construction possessive reduplication. The examples on Wikipedia may not match exactly the one the OP is asking for, but its meaning is clear and sometimes words have to be bent a little, to fit our needs.
Reduplication in linguistics is a morphological process in which the root or stem of a word (or part of it) or even the whole word is repeated exactly or with a slight change.
Reduplication is used in inflections to convey a grammatical function, such as plurality, intensification, etc., and in lexical derivation to create new words. It is often used when a speaker adopts a tone more "expressive" or figurative than ordinary speech and is also often, but not exclusively, iconic in meaning.
EDIT
The accepted meaning of "a man's man" is a man who enjoys men's activities and being with other men, i.e. the expression emphasizes the qualities of being a man. Likewise a lawyer's lawyers will imply that the person being compared personifies the typical or ideal lawyer. So perhaps the term emphatic reduplication or as suggested by @medica, intensifying reduplication, would be a more appropriate and accurate term.