Macs have the ability to display apps full screen and then switch between those different apps (usually using a 3-finger side-to-side swipe on the touchpad).
If the Teams app is full screen but is not the app that is on the screen then the notifications don't show up. The only way somebody would know that there have been any notifications would be on the badge in the dock, however if you've got the dock hidden (which I do) one might go a long time without seeing the badge.

Please could you make notifications show up regardless of which full screen app is currently visible. thx.

I want Preferences where it belongs, in the Application (Teams) menu, not hidden as "Settings" under my picture.
I want standard keyboard equivalents for menu items, like Preferences and Paste and Match Style.
Also hidden in the menu under my picture is a list of "Keyboard Shortcuts". Many (most?) of these items should also be menu items as that is what Mac users expect to make them learnable.
I want conversation controls to not be in the upper left corner where notifications (Mac or Teams) appear so I don't have to dismiss notifications in order to unmute myself. To be fair Apple puts controls in that same corner for Pages, Numbers, etc., but those apps aren't "live" where you are trying to ignore distractions.
I want standard Mac notifications, not a one-off solution that is often hidden by real Mac notifications.
I want real Mac windows, not colored frames based on which app I am running.
I want the app to actually resume where I left off when I switch back to it using command-tab, not just show a flashing insertion point in the message I was writing that ignores any typing until I click it.
I want support for standard emojis from the toolbar and other toolbar support.
I want better visual cues to indicate whether you are replying to an existing topic or starting a new one.
I want gif previews big enough to actually preview them without needing to insert them to see what they really look like.

To the Microsoft Teams development team,

Thank you for taking the time to read through these suggestions, and I am glad to be able to post my thoughts here in order to help create a better user experience. As a student, my experience using Microsoft Teams (specifically for Mac) has been mostly good. The app is very stable, and it runs well without using too much power. The functionality of having multiple office apps built in, with smooth integration with my One Drive makes using a single app for all my tasks a breeze.

However, I do have a few small additions that I would like to see. I often would like to be able to have my meeting full screen, but I also want to be able to access Team chats, private messages, and files open through the Teams app as well. The only problem is that I can only have one instance of Teams open at a time, in a single window. A fix to this problem that I see is being able to have multiple tabs or windows open in Microsoft Teams, thus allowing me to quickly switch between these functions of the app without losing changes in one section and retaining my full-screen meeting.
Another potential update that could be helpful, specifically for the Mac app, is better integration with Teams notifications with the rest of my system notifications. I would prefer that my notifications show up and stay in Notification Center until I view them; correspondingly, when I turn on Do Not Disturb, I don’t want Teams notifications to pop up either. The way that notifications within the Teams app populate two tabs at once (for example, being in Activity and Assignments) can be frustrating. I would like them to only be shown in one place at once. Also, integration with Outlook that allows emails and messages on Outlook to show up in Teams could be handy.
Regarding Teams video calls, I have almost no complaints. The ability to individually select my audio in/out and video input sources is quite convenient. The only thing I would like to be able to do is lower the adjust on Teams specifically.
These few relatively small additions would overall make my user-experience in the app smoother and easier, as well as allowing me to do what I want with the program. Thank you for your time and consideration.