Despite my username, I am a mathematician, and I can say doing mathematics definitely affected the way I think. Of course I do math as my job, but I also think of the rest of my life in very mathematical terms.
For instance, to measure my productivity I have created a detailed spreadsheet of my progress and the hours that I work. I view my efforts as a dynamical system, and potentially I can use this to clarify and understand the periodicity in my productivity using differential equations.
I view my purchases at the grocery store as an economic system and I have often come to rational decisions about money management using decision theory.
When I drive I think of minimizing the time of my route using traffic models. I probably haven't gained much on my travel time but looking at everything mathematically has clarified my view of the world.
My point is that every problem I encounter my mind can't help but look at it from a mathematical perspective, and the act of formulating problems in a precise way with all the necessary hypotheses have helped me solve many problems, even those that don't require heavy mathematical machinery. Mathematics isn't just solving specific problems but looking at a question from all perspectives and formulating thoughts in an extremely precise manner. These are things I of course did not do before I started to study mathematics.
The effects on me are pretty apparent because I have been doing mathematics for so long but I believe even a little bit can be very useful.