How do Terence Tao's "three stages of rigorousness" relate to cognitive science?
Mathematics is the


  abstract science of number, quantity, and space, either as abstract concepts (pure mathematics), or as applied to other disciplines such as physics and engineering (applied mathematics)


Cognitive sciences are 


  the interdisciplinary, scientific studies of the mind and its processes


I think that mathematics is a way to achieve something, to put it bluntly, it's a tool. A tool to, for example, model certain brain processes. Further,  math is an abstract tool; cognitive sciences is an applied science. Cognitive sciences can use math to accomplish its goals. By contrast, math can never use Cognitive Sciences to prove anything.

Now you say that rigor means 


  [an] emphasis [on] theory; comfortably manipulat[ing] abstract mathematical objects without focusing too much on what such objects actually “mean"


I don't think one can study the mind without focusing on what it means. Instead, studying the mind is often done exactly to try to find out how it works and what certain findings mean! Now - 

Scientific rigor in general means


  the quality of being believable or trustworthy


While in maths it has a specific meaning:


  logical validity or accuracy


So in terms of semantics the answer is that it is indeed a matter of meaning. Rigor can mean a lot of things, which is different across disciplines.