The short answer is yes, the Office 2021the licence is the same for Windows and Mac versions of Office; the long version is Macs need a little more work in order to get things going/up and running.
Some Quick fixes to try:
1. Use a Windows device, sign in with any non-Uni email and then apply the Key/redeem the key. If it works, you may be able to sign in on your Mac and use the licensed Office.
2. Uninstall any Office 365 from the Mac, including any free trial and then run Word and try to apply the Key.
4. Office 365, will/should just work, and if there is any way DSA/SLC will transfer the type over from a Key to 365, it will just work on the Mac. Ms Love that subscription...
If any issues still arise after attempting the quick fixes, apply the more in-depth fixes below.
If You have already partially redeemed the key/have assigned it to an email or tried using the key on a Mac that has Office 365 trial installed, or used a non-Microsoft email, it will now be a Microsoft support issue.
If not, there are things that can be done to get Office 20xx up and running.
TBH, they should be done anyway before contacting Microsoft.
Second, make sure you have an MS email. Hotmail, Outlook etc, it does not matter, as long as it is a Microsoft-provided email. if you get one drive etc, that is fine/that is the right email.
In most instances, you CANNOT/SHOULD NOT use a university-provided Microsoft email to redeem/sign up. Once Signed up, you can "link" the Uni email/have it as a sub email of the main subscription so you can access any Uni work, etc.
If it works and you sign in with your Mac, downloads should then be available, or at the least, the online versions of the app should be available.
Again, It cannot be a Student email if that has the "free" Office 365 license attached. The likelihood is the organisation/Uni have not set it up properly for Mac (15% of the user base, no school seem to set it up correctly).