The distinction here (which can be hard to grasp when overloaded words like "control" are used) is between trying to make population-wide estimates of effects of a factor, versus adjusting for a factor to get a better estimate of an average treatment effect.
As Mantel and Haenszel put it 60 years ago (page 729):
A sensible rule is to match those factors, such as age and sex, the effect of which may be conceded in advance and for which strong evidence is available from other sources, such as mortality data and morbidity surveys. When a factor is matched, however, it is eliminated as an independent study variable; it can be used only as a control on other factors.
For example, if you match cases and controls based on age, then it would be inappropriate to try to evaluate the population-wide influence of age on outcome from your matched cases.
That does not, however, prevent you from incorporating age as a covariate to adjust for influences of age on outcome within your sample. As perfect matching is seldom possible, such additional correction for covariates is typically wise. That's simply an additional way in which the matched factor is a "control on other factors." Such correction for covariates can even be useful in randomized prospective studies.
For more details and other perspectives, look at the following pages on this site, and their links:
What is the link between methods such as matching and statistically controlling for variables?
How do you “control” for a factor/variable?
ANCOVA in observational studies: what are the assumptions?
The last page listed above has a reference to a very useful overview by Schafer and Kang, who evaluated 9 different ways to try to determine average treatment effects from nonrandomized studies. They provided detailed explanations along with examples from a realistic simulated data set. If you haven't read this paper carefully already I strongly recommend it, as it goes into issues far beyond simple matching that might be important for your study.