There is no specific term, or at least no commonly understood term. (There is a very rare term "pentad" or "pentade". I'd never heard it until now, so don't use it.)
The phrase "Round numbers" is like "large numbers". There is no standard definition of "large number", but we all know that 10 is larger than 8, and 1000000000000 is a large number in most contexts. Similary 2000 is rounder than 2010.
We don't often talk about "round years" but if we did I'd guess it would usually mean 1990, 2000, 2010. These are numbers that have been rounded to the nearest 10.
If your doctor wants to be understood precisely then she can't use "round years" because this doesn't have a precise meaning. Most likely she would say "Have a test next year and then every five years" (the numerical value of the year doesn't matter, but the spacing does). If the numerical value of the years does matter, she could say "in the first and fifth year of each decade". (Although literally this is 2000, 2004, 2010 2014. So some longer description is needed. "at the start and in the middle of each decade" or something similar.)
The company is audited each year, but (twice a decade/every five years), an in-depth audit takes place. However the 2020 audit was cancelled due to COVID restrictions. So the next in-depth audit will look at the whole period from 2015 to 2025.