Why are there no varieties of synagogue?

Answer 1

There are - there are Orthodox, Reform, Liberal, Masorti (also known as Conservative) and others. In addition, there are Ashkenazi synagogues (with congregations chiefly made up of Jewish people of North and Eastern European ancestry), Sephardic synagogues (Spanish, Portuguese and North African) and Mizrahi synagogues (Middle Eastern).

Synagogues do not fall into architectural types, which is what I suspect you mean, simply because there is no religious law or restriction as to what a synagogue must look like or how it should be laid out, merely what it should contain within. As a result, older synagogues have tended to take inspiration from existing local religious buildings (so many resemble churches to an extent) or may be radically different in the case of modern buildings.