Me and your mother might have many things in common but I doubt that the reasons of calling it "Website installation" is one of them.
I would say that if you were familiar with the web technologies you would have seen the parallels.
When you type an address of a website or click on a link your browser would download the website data from a server, which is very similar to what happens when you tap "Install".
Once your website data is downloaded, the browser would execute the instructions in that data. Usually further resources would be downloaded and, UI drawn and functions executed. This is again very similar to any apps that you run once downloaded.
Installation of an app implies making it ready to use. You can download an executable that runs right away or you can have an executable that prepares the environment for use. This is again exactly the same with visiting a website, it might be ready to use right away or in might need a set-up of the environment.
In 2021, most websites need a setup process before use, you need to set up the ways you are about to be tracked and what information is to be shared with their partners. You need to decide if you like to sign up for newsletter, if you like to follow their social media accounts and more often than not you will be required to create an account. You will be prompted to set-up notifications too, if the browser support it.
Often the native App installation is less tedious and less painful than the website installation process because the app can afford to assume that the device is a personal one and the stored data wont be erased accidentally, therefore the account creation process can be both transparent and persistent.
With native apps, you get the chance to see the app before installation(App Store pages with description, screenshots and reviews), with websites you first need to finish the installation and then you get the chance to see the website.
Anyway, I hope you see how a website could be installed and in fact is the standart modus operandi in 2021.