Australian financial service (AFS) licensees need to quote their AFS licence number in certain documents (s912F Corporations Act 2001). The documents that this obligation applies to are outlined in regulation 7.6.01C of the Corporations Regulations, for example a Financial Services Guide.
Emails are not included in this list. However some licensees include the licence number in an email signature as a matter of good practise. Other times it might be included in the email signature for a corporate authorised representative or sub-authorised representative to make it clear that they act on behalf of the licensee. (This helps to avoid allegations of misleading and deceptive conduct).