Office keys are platform specific so, no, in general it won't work. Your license key for Windows won't work on Mac. If you got the software as part of an OEM bundle then this is more likely.
But you said you log into your MS account on the Mac and you see the option of downloading Office. You're doing that through a website so all it likely knows is that you have an Office license and not, at this point, that you're on a Mac. If it knows you have an Office license just based upon your MS account then it sounds like you might have a digital license. In that case you should be able to install it on the Mac as well. If it simply shows you a product key then you have a Windows-only license.
Of course this is easy to confirm, just select the option to install the Mac version. If it doesn't give you this option then it won't work. In my case I don't have an option to install for Mac. If you do have the option to install then install it. After installation it should try to authenticate. If your license is good then it will work. If it fails then you'll need to get a license. Note that an MS 365 Family Subscription is generally cheaper than buying a license for the product. So if you find you need to get a license then you might look into a subscription first.