If you want to check if the update is applied:- In the admin panel in the bottom right of the page it will tell you your Magento version (should be 2.3.4 for latest version)- Or you can run php bin/magento --version from the Magento root
The message that pops up on login is old (28th of Jan) and usually will say until you mark is as read / remove it. You can do this by login into the backend and going into "System > Other Settings > Notifications". And find the notification about the update and mark it as read / remove it.
EDIT
The relevent module that controls these updates notifications is Magento_ReleaseNotification. If you really want to disable these messages in the future then disable this module rather than Magento_AdminNotification.