UPDATE: On Sept. 10 I received notice that this UserVoice item will be implemented in the near future. See Syncing Email Signatures Across Devices is soon to become a reality!!
The current email signature behavior in Exchange and Exchange Online is that users must create new signatures every time they add a new mailbox in Outlook and in each of the Outlook apps. OWA already stores a text and HTML signature in the mailbox, but Outlook doesn't use it. OWA also does not provide for different signatures or a different reply signature like Outlook does. That's very inconsistent behavior.
In a recent discussion with the Exchange and Outlook product groups, the MVPs discussed a long-standing request - to store email signatures in the user's mailbox. Doing so will provide a centralized place to store and retrieve the signatures and provide consistency for the email clients that consume them (Outlook for desktop, Outlook on the web (OWA), and the Outlook apps (iOS and Android)). We are also requesting that signatures can be managed via PowerShell.
The product groups challenged us to show that customers want this by vote count on UserVoice. Please vote for the "Store Signatures in the mailbox" idea on UserVoice website to make your voice heard. I've written a spec for this feature, which I will be submitting to the PGs once the vote count gets higher. Our expectation is that this will work for both Exchange Server and Exchange Online.
Please up-vote this feature request and let's see if we can get this one done!