Shopify presents this error when the shipping from field does not match what the postal service has on file. You would get the error "Your store's shipping address couldn't be verified" when the physical address they have entered in Settings > Shipping into the Shipping from field does not match the information in Google and should look something along the lines of the screenshot below:
My guess would be the app is not verified or that there is something very small off with it, as simple as the postal code or one letter being lower or upper case, this is something which Shopify doesn't control and thus can't action a fix for. I spoke to our higher technical team who suggested to report the error to Google directly via the link here. Hopefully this works.
In particular, you need to enable the Geocoding API. This API is what converts and address into a geolocation.
Just to confirm if you have generated a new Google Maps key for this? If you haven't that could be the problem. Could you follow the steps in the help guide for Registering a Google Maps API Key here and then let me know if that works for you please?
I had a similar issue. After going through every single page, I discovered on Shopify's instructions to make sure you have your billing account linked with Google Service. This will still be under a trial basis and won't take money out of your account until you manually upgrade to this. Your account will have a $300 trial credit to use. Hope this works for you as it did for me.
I suggest you try the advice given above by @iamnotkiran. This is my personal account.
I had a similar issue and tried everything until I found the advice by shopify as their tips. By simply registering a card in payment / billing, not necessarily paying for anything solved this issue for me. Hope this helps.