Can I install the native Android OS on a non-Google phone?
I think you want to run AOSP like ROM on your device instead of OEM's bloated version. The answer to your question depends on the device you are using. There are a lot of custom ROMs available for devices. 

For finding these go to XDA and search for your device.
There you can find all the ROMs available. LineageOS (formerly known as CynogenMod) is the most popular one.