Welcome to community, If you could share the site URL it will be easier for members to look at and potentially give you a solution
Nick here from Shopify.
Can I ask what mobile browser you are using and if your phone is up to date as well as the browser app is fully up to date? That is generally the main reason why the Favicon wouldn't show up on mobile. I did a quick check on my iPhone browsers (Chrome and Safari) and it shows up for me. A Favicon would only show up on the switcher screens via mobile and not when the website is showing. I took a screenshot of each which you can see below:
Can you let me know if your phone and apps are up to date and see if they show up then, please?
All the best, Nick