More can be said but it was complicated.

Where did the Asians sit on the bus during the US segregation? written by u/EdHistory101 provides an excellent synthesis and lack of logic behind segregation. Additionally two other top level comments link to other threads

How were non-black minorities treated in the Jim Crow South? and If an Asian-American went to a diner in the Jim Crow South, where would they be seated? written by u/Dubstripsquads (note that these are the same text answer, however, there are some different follow ups)

Another answer of mine you may find useful is this one about Presidents and their schools. The relevant content:

We can also see this separation idea play out for Asian children on West Coast. Before 1885, there was an ad hoc system of schooling for Chinese children in San Francisco; parents got tutors, churches provided schooling, or some times the city would support a Chinese-only school. In 1884, Joseph and Mary Tape, two Chinese immigrants, enrolled their child in their neighborhood school and were denied. They took their case to the state Supreme Court and won. In 1907, the city tried to insist that the Japanese children in the city needed to attend the Chinese school, rather than a school with white children - even if the Japanese children spoke English. Parents raised their concerns with the Japanese government and shortly after, President Roosevelt got involved. The resolution to the problem was known as The Gentlemen's Agreement and helped contribute to the groundwork for Brown v. Board.