No, it is not a breakthrough case --- it is an unreported infection.
As noted many times before -- there are likely 2 to 3 estimated infections for each reported Covid case. This information is backed up by other types of testing such as sewage testing in many cities to determine overall infection level.
Early in the pandemic when there was not much testing available -- the ratio was likely 10 estimated infections for each reported case. But this has reduced over time -- in an current environment with ample testing -- to where it is just above 2 to 1 in the U.S. (as per late October estimates). It is likely increasing with Omicron -- but sewage monitoring still shows it below 3 to 1.
Note that it is likely that a greater percentage number of unreported infections are among the unvaccinated rather than the vaccinated -- following the trend of reported cases. This means from an overall "breakthrough scenario" the unreported vaccinated infections are less than those among the vaccinated.