Not every newly married couple has a Shinto ceremony, but it appears to be more frequent than before. Odds are that the couple will have worn a western wedding dress and suit and cut a wedding cake at the party at the hotel. Quite a lot of non-Japanese men make some money on the side by acting as a minister and performing a mock ceremony. There is no need to be ordained, in Japan one is married by filling out some paperwork and handing it in at the City Hall and any other ceremony is just ceremonial.
I do like the Shinto ceremony and, in fact, had a Shinto ceremony six months after I was married at the Shiogama jinja.