I am considering doing a PhD in a great lab/with a great advisor (and already convinced the PI). But I got in another program rather than the supposed program with the name of my subject of study, e.g. EE/chemical physics vs physics. I was worried as a rookie whether this degree would put me in an inferior position in the future competition. Regarding the circumstances in academic positions or industrial ones, how much does the degree name affect job hunting?
In many cases it would matter only a little if at all. Many job offerings call for a degree in "X or a closely related field". Physics and Hungarian Literature probably don't qualify, of course.
But in any case your application for a job consists of more than just your list of degrees. You have a chance to tailor your materials to the job at hand and so can say (must say) why you are appropriate for the job.
However, there are exceptions. Some countries, notably Germany, have laws relating to the required degree for a job. Some professions, notably medicine, have expectations that are rarely ignored.