There are people that can easily swap roles, such as going from a general hardware engineer to an IC design engineer. There are people who get out of school and immediately go into analog IC design. I know, I did that and have been doing it for almost three decades. From there, there are people who cannot make that switch either.
With all of that said, there is often much more difficulty in going from generalized roles to more specialized roles as opposed to the other direction. One of the beauties of having someone go from straight out of school into the IC design role is that the general hardware gets taught via mentoring at the same time as all of the other relevant skills. There is a LOT of additional specialized knowledge that has to be learned to go into IC design as opposed to general hardware design. Even within a field like analog IC design, there are specializations that go even further down.
There is a lot that needs to be covered in school to get someone ready for going into general analog hardware design. Starting of course is circuit basics such as RLC equations, transistor basics, building blocks like op amps, logic, etc. that all eventually go into specialized circuits like a switching supply or a signal acquisition board. To go into IC design, now you need to get deeper into transistor level circuitry, covering things like crystal structure, diffusions and device physics. Depending on the school you go to, you will find it easier or harder to get into these courses. Not only that, but the development of technology over the years has made it such that it now usually requires master's level study to get this education compared to decades prior.
Once you actually get out of school and into a role, understand that you don't just get turned loose to do the job. The rule of thumb we used is that it took five to seven years of mentoring before a new grad would be prepared to be working without major oversight. First projects were almost never full chip designs, but instead being given an existing circuit to make changes to in order to create a new product--imagine taking a 1A buck regulator and changing it to now be a 3A buck regulator or changing it to run at lower operating currents. This gives one a chance to learn transistor level building blocks, chip layout, and eventually circuit debug in the lab (which give the general hardware experience).
As for working at an analog IC design company, there are many different roles available. The IC designer is often the toughest role as they need to understand (to some degree) every other role, starting from process engineering at the base level, to IC design, to layout, to packaging, test, and finally application level. Each of those other roles ends up being a different person they interface with in the course of their duties.
If you have questions, let me know.