@cbrown3,
So what you’re seeing is the macOS “Handoff” feature and actually has nothing to do with either browser or either search engine. Basically, on your iPhone, you’re using a browser and the handoff feature detects this, and allows you to continue that work on your macOS using the default browser selected on the system — in your case, Chrome.
If you were to turn off the “Handoff” feature, you would no longer see this behavior.