Dear István HORVÁTH,
Good day!
Thank you for posting to Microsoft Community. We are glad to assist.
As per your mentioned description about "No 'SAVE AS' option for attachments in new Outlook desktop app". after thorough research and testing I found that this particular setting only exists in the classic Outlook and is not yet available in New Outlook for Windows at this moment. I have tested on my side and reproduced your issue as shown in the attached screenshots.

The New Outlook is in a Preview state, and they are still working on the platform/foundation of the application itself. At this stage of the development, luckily restarting the client is very quick. As an alternative, you can switch back to "Classic" Outlook by turning New Outlook Off

However, thank you so much for bringing this to our attention, your suggestion is very helpful, and I believe it could make the use of New Outlook even better. Since New Outlook for Windows is still in active development, we will surely raise your concerns to our Product Development Team and hopefully this can be included in the future updates. In the meantime, we sincerely recommend you post your concern as an idea to our Outlook in-app feedback site Contact support and provide feedback within the new Outlook for Windows - Microsoft Support

Our Product Development Team are committed to enhancing the customer experience by continually examining suggestions to include in the design and development of future product upgrades based on your suggestions. Hopefully, your suggestion will help many users in the future.
Thank you for your time and understanding. For other questions, feel free to add your post in the Microsoft Community. Have a nice day!
Best regards,
Inema | Microsoft Community Moderator