I am having the same issue on my Samsung Galaxy S7 with the latest OTA updates (Android 8.0.0) and latest LastPass update from Play Store (126.96.36.19914) Built 8 June 2020, 7:59 am and Developer Settings Enabled.
The notification says, "to use app fill on your device, enable the 'draw over other apps' permission for LastPass."
The "GO TO SETTINGS" link in this notification goes to a screen labeled "APPS THAT CAN APPEAR ON TOP", with only LastPass shown and the "Allow permission" slider turned on.
I found another way to get to a similar screen, which shows this permission for LastPass and many other apps. This is Settings, Apps, upper right vertical ellipsis, "Special Access", "Apps that can appear on top". The slider for LastPass is also shown enabled on this screen.
Thu, 30 Jul 2020, 10:37 am EDT (UTC -0400)