Product design is quite a broad field obviously, first the best thing to do is look at what the alumni of the university end up doing, and what are the current research/project interests of the department?
The large fields of product design can be things like furniture design, footware design, sports equipment design, consumer product design, and industrial design, with subsets for things like vehicle and automotive design.
I am a mechanical engineer by degree but take on product design projects as a consultant, there are a number of skill sets to build which your course will give you, to varying degree's based on their strengths.
Firstly understanding that design is about solving problems with the tools, people, and resources you have available. Being able to communicate your potential solution is a key skill.
This is manifest as sketching and drawing, both on paper and digitally, you do not have to be an artist but as a product designer being able to quickly develop concepts visually is important for you to refine your ideas and to communicate to the client. - as an un-artistic engineer I am not the best at this bit on paper, but I am a very good CAD (computer aided design) software user ( see solidworks etc.)
Secondly understanding "some" of the physical principles that might underpin your design - this bit comes very easy to me as I have the technical background but understanding the limitations of your materials, basic concepts such as density, stiffness, strength, toughness, and what things might be important for the design brief (oh a high temperature environment - better understand if my material will suffer from corrosion and thermal expansion etc.)
Also manufacturing methods - many less industrial product designers have no clue how to actually make the thing they have designed, this is problematic as design for manufacture is an extremely important thing for viability, cost feasibility, and scalability of your design, which is often neglected.
Thirdly - Understanding the clients needs and being able to capture the true design intent of the brief - this is helped by various frameworks such as design thinking.
As for the sort of jobs you can get there are product designers in virtually every company, even service based ones use designers to improve their digital product designs.
Understanding the nuances of design and being able to produce, and recognize good design, is a trait that is extremely useful for all companies.
So in a nutshell it depends what you would like to do, do you want to make consumer products, or industrial ones?
Digital things or physical things? Cars, or trains, or boats, or medical devices. Maybe Trainers or tennis rackets.
The first thing I designed was mechanisms and components for circuit breakers, the last thing I design was motorcycle parts, in between I've designed things in automotive, electrical, domestic products, adapted devices for the disabled, and many more.