All my experience in academia in western countries is that if you're an enrolled student or affiliated faculty, you get an email address from that university. Are there countries where this is not common practice? And if so, is there a good reason for not providing one?
Many universities, especially private universities, in India do not provide an email address to its students. This is due to the cost of maintaining email server, ensuring confidentiality etc. Even in top universities in India such as the IITs, where everyone is provided with an email address, the email server is suspect and fails.
While I was teaching a graduate class, I circulated a sheet in which they were supposed to write down their email address. I would then make a group and circulate any class related notes or ideas. I noticed that 90% of the students gave gmail or yahoo address and not the institute email address. When I inquired further, I was told they never had an institutional email address at their undergraduate university and are not accustomed to it.
Many faculty in Indian universities also do not use official email address due to reliability problems. You can notice this from noticing their publications. The address of the corresponding author refers to an university while the email address is generally yahoo/gmail.
No, it's not standard.
It's the same as with all institutions everywhere - if you are employed by that institution, you get an email address that makes it clear you are affiliated with that institution. This means you can send emails that immediately identify you as a member of that institution. Once you leave the institution, you will also lose access to that email (or you will get another different email address), which again makes it clear that you are no longer a member of that institution. This has obvious advantages such as making it harder to spoof others.
This applies for not just universities, but also companies big and small.