How many shrines outside Japan? 666 in 1945, just a handful now
From 1896 the Shintō Shi DaiJiten gives numbers for shrines outside Japan. Unlike the faith/dogma-driven two newer Peoples of the Book there seems to have been relatively little proselytization on the part of Shintō, it was very much a case of following the flag: Taiwan was ceded to Japan in 1895 and in 1896 the first Shintō shrine was dedicated there. The 1904-1905 Russo-Japanese war was essentially fought to decide who would control Korea and Manchuria. Following Japan’s costly victory the US-brokered treaty signed in September 1905 not only formally acknowledged Japan’s dominance in these two areas but also resulted in the southern half of the island of Sakhalin being ceded to Japan by Russia. The first Shintō shrine in Manchuria was dedicated in 1905, three shrines were established in Sakhalin in 1910.
This table shows the first shrine in Korea being set up in 1915 but there were already shrines in Korea by that time. In 1876 the Japanese-Korea Treaty of Amity was signed. Almost certainly modelled on the unequal treaties forced on Japan following the visit of Commodore Perry and his black ships it, among other things, ended the Chinese protectorate in Korea and gave extraterritorial rights to Japanese citizens.
There is an excellent English language study of these shrines outside Japan by Nakajima Michio, “Shintō Deities that Crossed the Sea,” in the Japanese Journal of Religious Studies. The table Nakajima uses is essentially the same as the one in in the Shintō Dai Jiten and gives 1911 as the year the first shrine was set up in Korea. In his text, however, he notes the erection of two shrines by Japanese migrants to Korea, one in Wonsan in 1882, the other in Inchon in 1886. He gives these as examples, the implications being that there were others.
As it is almost certainly nothing more than a coincidence, pondering the relationship between the non-Japan 666 shrines (the table shows 655 but there were a further eleven undated ones) and Revelations 13 is not an issue I am interested in exploring: I will leave that to the reader.
Note that the table refers only to areas which came under the sphere of direct Japanese influence. There are a handful in other areas, particularly Hawaii and Brazil, which followed Japanese emigration to those countries. The Green Shintō lists seven for Hawaii and nine for Brazil. The same site also lists one shrine in each of Holland, France, the US, and Canada.