One way that I was able to find the cpu architecture for my google Pixel was by going to "settings/Apps/See all apps/Clock/App details". That will get you to "Google Play". On that screen below "Installed on all devices" I see "Google SDK_gphone64_x86_64" and it says its "Installed". So you can then infer that my phone architecture is x86_64. I also assume that you can infer that this is a 64 bit architecture (gphone64). So that's one way you can get the architecture without any special application installed or using an "adb shell".
Please note that I'm not sure this exact method will work for all vendor Androids on the market. You may need to look at the "App details" for some app other than clock. So substitute another app's name for clock in the settings path I described above.
It's too bad that the phones cpu architecture is not listed in "settings/system" or in " settings/About phone".
Update: Based on the comments from John and Andrew, I'm going to say that my answer is wrong. This doesn't work as a method for inferring the processor type without the aid of some other tool as mentioned in the other answers. Apparently, the App Store keeps information about all the apps installed on all your devices just by the fact that you visited the App Store. I must have visited the App Store from an Android emulator I was running, and the "app detail" from the App Store confused me. I was hoping that I might find the name of a native library code installed with some app by looking at the "app detail". Apparently that's not the case.