This is some sort of sports team, from the Washington State Historical Society:
But, I really doubt it is a basketball team, as labelled by the WSHS. Mostly because it labelled as being taken in 1885 and James Naismith didn’t create the game until six years later. I think its much more likely that what we have here is an actual soccer team.
For one thing, the year is pretty good for the spread of the game. The first nationwide soccer association launched in 1884, the first national cup in 1885 along with the first international friendly. While all three of these events occurred inside the New York/New Jersey area, soccer obviously existed while basketball did not.
You can find some trace evidence of soccer in Washington State in the same era. This 1891 newsclip from Yakima mentions soccer being played.
Now, here’s some funny history about the Olympia Collegiate Institute. At different times, appently both the Methodists and Lutherans ran the old OCI, but merged them with Tacoma-area schools at different times. The Methodists absorbed OCI into the University of Puget Sound in the 1880s. The Lutherans restarted OCI (Later the Pacific Lutheran Seminary), absorbing it into Pacific Luthern University in the 1910s.
While Tacoma couldn’t end up stealing the capital from Olympia, they did make away with two colleges.