Ian's hair appears to be different lengths in out of sequence scenes, indicating either the original filming schedule or that some scenes required pickups.
Judging from the nearly overhead sunlight, the wedding is held during the afternoon, which is normal, but the high school prom is happening at the same time, which is odd. A prom in Chicago in the first half of May would typically start about the same time as sunset (7:30-8pm), so if the wedding started at the same time, it should not have been broad daylight when they exited the church.
At the beginning of the movie there is frost on the car and all the leaves are dead on the ground. This indicates it's mid-late fall/early winter when Paris is looking at colleges which is aligned with January deadlines most colleges have for the following fall. The movie skipped over winter to late spring.
Paris is shown applying to colleges a few weeks before graduating from high school, yet she is accepted to two schools who have application deadlines of January 1st. The type of school College Fair she is seen attending in April and May is intended for juniors, but she was a senior.
In the first movie, Toula and Ian marry when she is thirty two and her sister Athena is thirty five (three years older). The second movie is eighteen years later, which would make Toula fifty and Athena fifty three. Yet, repeatedly, their parents, Gus and Maria, are said to have married "fifty years ago".
Gus' car has a license plate that says OPA! It is not possible to have anything other than numbers or letters on a license plate.
Gus is convinced that he is a descendant of Alexander the Great. However Alexander's only known child a son also named Alexander died when he was 14.
In Orthodox Church, is forbidden for a female to go behind the "Templo". This rule is called "Avaton". In the scene of the wedding, Maria goes there (when she doubts if she should continue the ceremony) and the rest of the girls of family follow her to change her mind. The priest should have made a reprimand to them. Also because they follow the ethics of Greek culture, this should be known to the characters without doubt.
When Greeks makes a toast, never say "OPA". They say "YIA MAS", meaning "Cheers" or, more literate, "have a good health". "OPA" is something Greeks say when they dance.