Now that Google closed their Chrome webstore payment service, VDH licenses, whether they are for Firefox, Chrome or Edge, are exclusively purchased from the
https://www.downloadhelper.net/convert page. As you noticed, the web site detects the browser and offers to purchase the corresponding license.
The idea is that when you purchase the license, a new license ID is generated and automatically installed in the current browser. Of course, there are ways to install this license ID manually but it requires following a couple of links in the extension and performing a copy-paste operation which might be a bit complex for non-technical users. In practice, the automatic license install is what's being used in the vast majority of cases.
I am unsure if use-cases of purchasing, say a Chrome license from Firefox, hence preventing the automatic install, are frequent enough to take the risk of misleading the user into purchasing the wrong license type by providing cross-browsers links.
By the way, the browser-detection-bypassing links you are looking for are: