The GitHub advice is possibly over stating the risk. Using the same ssh key on multiple systems has nothing like the risks associated with having the same password on multiple systems. However, there are some other factors to consider which may result in someone having more than one ssh key.
The most obvious reason to have more than one ssh key relates to your concept of identity. While most of us consider we only have a single identity, the reality is that we probably have at least two - a personal/private identity and a work identity. For most of us, our personal identity never changes, but our work/professional identity could change many times during a career. Therefore, it often makes sense to keep these two identities separate. In some organisations, you are even issued with a 'work' ssh key and when you leave the organisation, that key is revoked. If you use that key for personal purposes, you could find you have lost access to some resources or have to go through a vetting process to regain access etc.
Another reason to have more than one ssh key is to limit the value of your ssh key. This is similar to having two separate key rings - one which you hang on the hook in the hall and the other you keep in a safe under your floor. The first contains the key to low risk, frequently used resources. The second, high risk or seldom required resources. If your house is robbed, they will likely get the first, but not the second, minimising the damage.
This last item relates to key management and in my experience, the most frequent weakness/failure I see with respect to ssh keys and one of the reasons I think we have not seen wider deployment of ssh keys. I frequently encounter situations where people are using an ssh key, but have vary poor key management practices. The reality is, using ssh keys with poor key management practices is often less secure than relying on just passwords. Many users don't understand how critical it is to protect their private key and will make mistakes, like checking them into version control systems, storing them in unsecured backups, copying them to various cloud storage systems etc. Too often, I find people using ssh keys without a passphrase because it is more convenient. This is a vary bad idea. If you want convenience, then use an appropriately configured ssh agent, but make sure you have a good pass phrase (note the use of phrase rather than word!).
It is because of this key management issue I would not recommend using different keys for every site. Apart from not improving security, it adds to the key management burden - the more keys you have to manage the more likely mistakes will occur. Most people will likely want more than one key, but it would be unusual to need more than 5 and in most cases, 2 or possibly 3 would be sufficient.