You can't just "go from ethernet to WiFi" because these are two different types of signal. Anything that advertises itself as an "adapter" is known as a Wireless Access Point (WAP). These are made to connect to existing wireless networks. That said, they are made for existing wireless networks to which you have access to add Access Points.
Please note that the little device for a VAIO you have in that link isn't an exception to this rule. It's just a small one that uses a VAIO power adapter. Beyond that, it's not different from any other. WAPs and routers with a built-in WAP don't care who made your computer.
Also note that adding a WAP isn't like just connecting your phone to Starbucks Wi-Fi. It usually requires access to the network as an administrator and a little bit of setup to make sure it's broadcasting on the same SID and, in the case of a portable router, is passing traffic and DHCP assignment management back to the router that actually runs the show. For that reason, a product like the one in your link will probably end up being a useless box of plastic, as taking it with you will still not enable connection to secure Wi-Fi in may cases.
If your goal was consuming Wi-Fi, I'd say get a USB wireless NIC. Since your goal is to broadcast it as a hot spot, you'll find that this can work a couple of ways. First, Windows 10 has the ability to broadcast as a hot-spot using the existing hardware or a NIC like I just mentioned. Your computer serves as the host. Second, see what it will cost to use your phone as a hot-spot for your wireless carrier. This is usually a per-megabit charge or an additional charge at a flat rate added to your bill to give your phone permission to do so.
If what you really need is to connect multiple machines to wireless and allow them to connect to each other directly across that network, you'll need the WAP, but don't expect everywhere you go to accept your connection.