So it is my understanding the Washington's Olympic Mountains are basaltic mountains. The reason is that they formed from the upthrust of Ocean crust rather than continental crust as the Farallon plate was pushed under the North American plate. Are there any other mountain ranges (worldwide), particularly coastal ranges made of basalt oceanic crust, that have roughly the same geology?
Short answer: yes, there are other mountains which contains oceanic crust in their lithologies.
The process you described as "the upthrust of ocean crust" is called obduction. The resulting rock sequence is called an ophiolite. As you can see here, there is quite a lot of them around the world! One of the most studied is the Oman ophiolite. I have seen the Chenaillet ophiolite in the French Alps, it's quite spectacular.
Note that ophiolites usually include not only basalts (often in form of pillow lavas, typical of emplacement under the ocean), but also other oceanic crust lithologies such as gabbros, and even the underlying lithospheric mantle, often as serpentinite (a peridotite altered by water).
I found a Guide to the Geology of Olympic National Park on the USGS website. Unfortunately some links are dead, but you can still find lots of information on the range's lithology and formation. The page on "basalt and its associated rocks" mentions the presence of gabbro (a magmatic rock with the same composition as basalt but a crystalline texture due to slower cooling), but not that of any mantle rock. According to this other page on "colliding plates and Olympic rocks", the basalt is not from the oceanic crust itself but rather from a submarine volcano emplaced avove the plate (fig. 26). So, the Olympic Mountains would not be an ophiolite per se, but rather an accretionary wedge, i.e. "a thick wedge of rocks produced by the progressive offscraping of ocean floor sediments during subduction and their accretion to the continental margin" (excerpt from the guide's preface).