Does mean solar time and sidereal time sometimes indicate the same time?
The 3m56s mean difference between solar and sidereal days is due to the Earth's orbital motion around the Sun.
This adds up to 1 day per year; for every 365&frac14; solar days there are 366&frac14; sidereal days.

Sidereal time equals the right ascension of what's on the celestial meridian at that time.
The Sun is on the meridian at 12:00 apparent solar time.
These align at the September equinox, when the Sun is at RA 12h.
Mean solar time is ~7.5 minutes behind apparent solar time on that date and aligns with sidereal time ~1.9 days earlier.