Safari search engine always resorts to google search engine on it own after changing to other search engine

Thanks, that information rules out a lot of other possibilities.

If your description occurred while using that Mac in "Safe Mode", and it persists after a normal restart, and given everything else you did leading up to that point, I have nothing to suggest other than to reinstall macOS. Something went wrong with its installation that I do not know how to fix.

Reinstall macOS - Apple Support

Reinstalling is nondestructive but you should create a backup anyway, as a fundamental prerequisite.

Or, contact Apple at your convenience: Contact Apple Support.

Either way please let me know what you determine.

I understand your selected Search Engine Preference is being ignored and that the following is not working as described: Change Search settings in Safari on Mac - Apple Support

Is it possible that you are using a Google Chrome Profile as described in the following Discussion:

Google Chrome Repeatedly Adds itself to Login Items - Apple Community

If not, does a Profile pane appear in System Preferences? If so it would look like this:

If you do not see one, please check to be sure it has not been "hidden" by selecting View > Customize.

I understand that Safari is reverting back to its default Search Engine, but only after you quit Safari.

Quick test: Please determine if the same problems occur in "Safe Mode": How to use safe mode on your Mac - Apple Support. Describe what happens when you launch Safari, change its Search Engine, perform a search confirming that change was effected, then quit Safari as you have been doing. Launch Safari again and determine if your Search Engine remained the same or if it reverted to its default again.

Then, restart your Mac normally and determine if the problem remains.

I may have more suggestions pending the results of your observations.