This is caused by chrome/android/java/src/org/chromium/chrome/browser/signin/AccountIdProvider.java doing the check for Play Services like this:

 63         UserRecoverableErrorHandler errorHandler = activity != null                                  
 64                 ? new UserRecoverableErrorHandler.ModalDialog(activity)
 65                 : new UserRecoverableErrorHandler.SystemNotification();
 66         return ExternalAuthUtils.getInstance().canUseGooglePlayServices(ctx, errorHandler);

Rather than using new UserRecoverableErrorHandler.Silent() like other code using Play Services.

Would it be acceptable to simply replace it? Spamming the notification doesn't actually tell the user which feature is unavailable so it's not a useful form of UX. It would make sense to show a dialog within Chromium when attempting to access an unavailable feature like signing in. Alternatively, the features that are unavailable could simply be hidden. It's up to you if you want to advertise that there are missing features they could have, but the notifications are just an annoyance and are incorrect since Chromium does work fine without Play Services now.

https://codereview.chromium.org/1553713002

The following revision refers to this bug:
  https://chromium.googlesource.com/chromium/src.git/+/7cf072a34e50d8cee9730f7d596c9a6f0a5cec8c

commit 7cf072a34e50d8cee9730f7d596c9a6f0a5cec8c
Author: knn <knn@chromium.org>
Date: Wed Jan 06 14:14:02 2016

Notify the lack of Google Play Services only if the user tries to Signin.

We do not show the annoying play services missing notification unless it
is actually required as a result of some action taken by the user like
signin.

BUG= 569285 

Review URL: https://codereview.chromium.org/1554103002

Cr-Commit-Position: refs/heads/master@{#367819}

[modify] http://crrev.com/7cf072a34e50d8cee9730f7d596c9a6f0a5cec8c/chrome/android/java/src/org/chromium/chrome/browser/signin/AccountIdProvider.java
[modify] http://crrev.com/7cf072a34e50d8cee9730f7d596c9a6f0a5cec8c/chrome/android/java/src/org/chromium/chrome/browser/signin/AccountTrackerService.java
[modify] http://crrev.com/7cf072a34e50d8cee9730f7d596c9a6f0a5cec8c/chrome/android/java/src/org/chromium/chrome/browser/signin/SigninManager.java
[modify] http://crrev.com/7cf072a34e50d8cee9730f7d596c9a6f0a5cec8c/chrome/android/javatests/src/org/chromium/chrome/browser/signin/OAuth2TokenServiceIntegrationTest.java
[modify] http://crrev.com/7cf072a34e50d8cee9730f7d596c9a6f0a5cec8c/chrome/test/android/javatests/src/org/chromium/chrome/test/util/browser/signin/SigninTestUtil.java

Should be fixed now.