Is it possible to have out of stock item to be sold to a customer tag of ‘stock’, but to everyone else, the item is out of stock and can’t be ordered?
Sure, you’ll need to allow continue selling when out of stock, you can then tag a customer, and use a checkout validation rule in an app like Blockify Checkout to disallow all purchases of that product if customer doesn’t have that tag.
Trying to hide add to cart button based on customer tag isn’t going to work if your website doesn’t recognize the customer when they are viewing the product page.
If you’re looking to make these products permanently visible (but unavailable) to a subset of users, you can use a memberships app like Zendra to setup restrictions. This allows you to show the product to all users (and, therefore, to search engines!) while also ensuring that only a subset of members are able to actually purchase that product.
You can even use that as an opportunity to get people to sign up for your email list or even purchase a membership from your store, depending on your usecase 
Just to build on what’s already been suggested - the two approaches aren’t really mutually exclusive, and combining them actually gives you the cleanest result.
Since customer tags can only be read when someone is logged in, that’s already a natural gate. You can use Liquid in your theme to check the customer’s tag and conditionally render either the Add to Cart button or a disabled ““Sold Out”” state. Logged-in customers with the ‘stock’ tag see the button; everyone else (including visitors who aren’t logged in) just sees it as out of stock. That handles the storefront experience.
The checkout validation layer Maximus3 mentioned is then just a safety net - someone could theoretically add a product directly without going through the product page, so blocking at checkout closes that gap. Without any frontend change though, non-tagged customers can still add the item to their cart and only get blocked when they try to check out, which isn’t great UX.
The memberships route kestrel-ian mentioned is worth considering if you want this to be a more permanent setup or if SEO visibility matters, but for a simple tagged-customer use case, the Liquid + checkout validation combo is probably the more straightforward path.
This is exactly what I would suggest, no need for any app just a simple smooth liquid code update, Good one!