Hello Damir,
At this time, sending a message from an email alias is designed to show the alias username as the display name. To differentiate which user sent the message from the alias, Domain Administrators can allow users to customize their signature mapping per alias. Otherwise, they can force a specific signature to be displayed. That said, I can understand your desire to show the display name of the user who sent the message instead. Or perhaps to even allow the Domain Administrator to configure a custom display name per alias.
Because this functionality is currently working as designed, I'm going to change this thread from a Problem to a Proposed Idea. For others who would also like to see this functionality, please be sure to vote up the thread and add your input on what you'd like implemented.
Thank you!