Note that Shanghai wasn't controlled by the Chinese authorities until 1946. Until then, much of the city was controlled by various foreign powers, chiefly the French, British, Americans, and, later, the Japanese. This state of affairs was because Shanghai had been ceded as a treaty port in the Treaty of Nanking at the end of the First Opium War in 1842.

Because of this unique legal structure, very little documentation was required to enter the city (and even those were abolished after the outbreak of the war). One of the rights of the foreign powers in the treaty port was the control over entry visas. Since Shanghai and the other treaty ports were set up to be trading cities, visa policies were very liberal.

After Pearl Harbor, the main International Settlement ended as the Japanese asserted full control over the city (the neighboring French concession, under the control of Vichy authorities, soldiered on until 1943). However, Jews were initially not treated very differently than the rest of the foreign non-belligerent population (that is, citizens of countries that Japan was not at war with - those citizens of countries it was at war with were interned). It wasn't until concerted pressure by the Nazis on Japan that they began to implement additional, punitive restrictions on the Jewish population, though the Japanese never took seriously the idea of liquidating the Jews (unlike their treatment of the Chinese elsewhere during the war).

Altogether there were about 20,000 Jews who escaped the reaches of Nazis by sheltering in Shanghai. These tended to be wealthier Jews as tickets to Shanghai were very expensive at the time and after the war ended almost all of them left, either to return to their home country or to immigrate to America.

Source: Port of Last Resort: The Diaspora Communities of Shanghai by Marcia Ristaino