I think the deeper question (aside from some confusion what exactly Outlook / Microsoft Exchange is) is whether it is ok for an instruction to insist on receiving emails from the university-assigned email address rather than some private address (independently of whether that is Hotmail, GMail, or anything else).
The answer to this question will, as always, depend on published policy by the university. However, in the dark I would assume that instructors are in the right here. Normally, universities assign students email addresses under the assumption that this is the way how instructors and other staff communicate with students. If individual students prefer some other medium, or a different address, this is something that you can take into account as an instructor, but I would be surprised if the university explicitly mandates doing so.
As somebody teaching in a university where a lot of students are in the habit of using private email for course-related communication, I can attest that for the instructor this is often indeed rather inconvenient:
- Private email addresses often use aliases that are not trivial to match to students on record, especially if the student is not from the same cultural region. This can be as harmless (but confusing) as students using a nickname that would require some detective work to match back to my records, but I have also received course inquiries from "Assbuster92" which, as you can imagine, is something I'd rather avoid.
- An instructor may specifically insist on communicating via the official email address when discussing sensitive matters, such as grades. Frankly, for a private GMail account, I have no way of assessing whether I am currently sending private exam data to the person I think I am sending it to. GDPR may also play into this - I am not even allowed to send exams and other sensitive information to a server not authorized by the university, even if I had somehow verified that the owner of this GMail account is indeed the student in question.
- For communication from me to the students (e.g., announcements), using anything but the official email address is very cumbersome or simply impossible. I don't have a list of privat student email addresses, and manually keeping track of which emails students use is frankly not a good use of my time in a class of any non-trivial size.
- If I send students emails to their university address, but they send me questions from their private address, communication tends to get scattered, making it difficult for me to keep track of what we discussed / agreed.
- Lastly, separating private and "work" life is a good habit that students should, in my opinion, adopt sooner rather than later. In any job they will work in the future, employers will also require them to use whatever email address they get assigned, even if said email address is not convenient to use for some reason. Forwarding work email to a private cloud service is strictly forbidden in any company with mature IT practices, so this is again not a habit that students should pick up.