Whether or not astrologers have conducted experiments has no bearing on the argument’s reasoning. The assumption is that the two individuals share a horoscope.

The psychologist claims that astrology does not determine personality, so another explanation of what does determine it (cultural differences) would potentially support the conclusion. The psychologist does not need to assume that this isn’t the case.

This must be true in order for the conclusion to follow. If the two individuals have different horoscopes, then this example no longer works as support for the conclusion that horoscopes do not determine personalities.

The psychologist establishes that the two individuals were born at the same time; information about other people’s births is irrelevant.

This is irrelevant. The only claim being put forward is that horoscopes don’t determine personalities, so it doesn’t matter whether twins have the same home environments or not.