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I recently noticed that when I go to manually search for my website, I'm able to find it no problem but the site title reads as "My WordPress"
I attempted to changed this to the right name by going to Settings > General > Site Title/Tagline. However, when I save and recheck, I still see "My WordPress". I cleared my cache and everything, retried the same steps as before and still no change.
Does anyone know how to fix this issue?

Could your theme settings be overriding that setting?
What is this a screenshot of? Where exactly are you searching? Google?
The text in your screenshot looks like navigation. Have you checked Appearance > Menus?