The answer is that, what a typical day for an ME is based on what type of job you get in Mechanical Engineering. I am in product development, but there are jobs in manufacturing, sales support, field service, HVAC, etc. I also work form companies that bring product from concept to production.
For product development, the daily routine changes over the course of the project. In the early part of a project, it is about getting product definitions and requirements. From their it moves to proof of concept, where you try to determine which path is feasible and meets the product needs. Then we get into determining the architecture of the product, how it goes together. Throughout we work with stakeholders in marketing, manufacturing, and service to align on their needs. Then we start into part design, prototyping, refinement of the design, pilot runs, and production ramp up.
I said a lot in very few words. The cycle can take from 10 months to 5 years for products that I have been involved with. What I like about it is that as we go through the process, the daily routine changes. We are always learning as we are applying new technology all the time.