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On Chrome for Android the menu usually shows either Install app or Add to Home screen — not both in one place. If you see Install app, use that for a real installed window.
If this site becomes installable, a blue Install app bar may appear at the bottom after you have been on the page for a little while — use that when it shows.
If the menu only offers Add to Home screen and the icon keeps a small Chrome logo, Chrome is treating it as a shortcut. Try again after a full page reload, HTTPS on default port (443), and ensure the device can install web apps (storage / Play services). Technical background: Chrome install criteria, shortcut vs install.