Every employer, including universities, should create email accounts for their employees/students to use for official purposes related to their jobs/studies.
There are many reasons that using private emails for these purposes can be a problem. Depending on what country you are in, these can include government records requests laws that would allow the public access to your private communications if you're using that private email for public purposes (in practice, this isn't quite as dire, but it's potentially a massive pain).
Not sure why your institution has not provided you an email address, but this seems very unusual. It isn't that uncommon for your institutional email to be used to authenticate that you are indeed the person you say you are and that you have some association with that institution.
I won't go so far as to say it's "absolutely necessary" but I'm having a hard time imagining doing any of my own work without an institutional email, and I would find it very very weird to work somewhere that wouldn't give me an email address.