I disabled cookies and I couldn’t get anything to work, so I enabled them again.
I carried out the test you gave:
I select product #1 and add it to the basket with the add-to-cart button in the description. I go to the basket, the product is added.
I return to the catalog and I select product #2 and I add it to the basket with the add-to-cart link in the description. Product #2 is not in the basket.
For me, that was only partly true. On the basket page, Product #2 is not in the table of products. Of course it should be there. But in the very top row of the page you have a mini-cart. There, the basket value and the number of products is updated correctly. And if you proceed to checkout, the order review table shows the correct products.
So it looks like the problem could be rephrased as the basket page table does not update, even though in the background the cart is being correctly updated and shows correctly elsewhere.
Next I think you should look at the template for the basket page.
Have you got any functions in functions.php for your child theme that may be affecting the basket page?
Please can you have a look at Admin page > WooCommerce > System Status > Template Overrides row, are there any template overrides that might be affecting the basket page?
Could you try the test with the parent theme activated, and also with the default TwentyFourteen theme, and see if the problem remains?