Both @TravisEz13 and @limonte have reported that the Grammarly extension is having issues with brave-core (reported here)
@TravisEz13 reports seeing this:

@limonte reports seeing this (on github.com):

Extension should work
BraveDev latest (Version 0.55.3 Chromium: 70.0.3538.12 (Official Build) dev (64-bit))
Is the issue reproducible on the latest version of Chrome?
unknown
I mentioned in https://github.com/brave/browser-laptop/issues/12289#issuecomment-421457707 that the same extension works with latest Chrome and Opera.
The issue is still presented in the latest relese of a stable channel (0.55.18)
Has anyone been able to log into Grammarly Premium and have the browser extension recognize it? If I go to app.grammarly.com it shows I am logged in. If I open the extension it says "Grammarly is active, but key features are missing" because I am not logged in.
I have not tried turn shields off. I'm not sure how shields effect browser extensions compared to websites.
BraveDev latest (Version 0.57.2 Chromium: 70.0.3538.67 (Official Build) dev (64-bit))
@christianjuth same problem here.
0.55.20 Chromium: 70.0.3538.67 (Official Build) (64-bit)
+1
+1
Guys, please stop spamming subscribers of this issue with "+1" comments, use GitHub reactions instead.
I got it to work.
In short, I think all you need is to click the login link and then restart the browser.
All the steps I have taken:
macOS Mojave 10.14, Version 0.56.12 Chromium: 70.0.3538.77 (Official Build) (64-bit)
I just noticed that without any additional actions (except keeping Brave up-to-date of course), the Grammarly extension started to work for me 馃帀
Windows, Version 0.56.12 Chromium: 70.0.3538.77 (Official Build) (64-bit)
I have Grammarly working on Brave Dev. I am also running a few other extensions (DuckDuckGoEssentials, Dashlane, and Honey) with no issues.
Mac, Version 0.57.4 Chromium: 71.0.3578.31 (Official Build) dev (64-bit)
It was working for me on the dev build, but I quit using for reasons not relevant for this issue. I'm fine if this issue is closed.
All known issues should be resolved now
Most helpful comment
I mentioned in https://github.com/brave/browser-laptop/issues/12289#issuecomment-421457707 that the same extension works with latest Chrome and Opera.