Paste this code in your theme.js in the head area.
<link rel="shortcut icon" href="https://cdn.shopify.com/s/files/1/0593/9855/3749/files/favv.png" />
Did you make any changes to get this fixed? I ask because I checked on both mobile Chrome and Safari and the Favicon showed up on both for me:
If it's still not showing up on your end, I'm thinking it could be a cache issue and it might be worth clearing your cache as both Chrome and Safari are supported by Shopify. You can see how to clear your cache for any and all browsers and devices here. Let me know!
I'm not 100% sure what you mean by the favicon not showing up on Google index. Could you elaborate a little more on that, please? The items Google would be searching for when indexing are the following: