I am trying to add the following code:
<SharePointWebControls:UserField FieldName="Site Owner" runat="server"></SharePointWebControls:UserField>
in a Script Editor Web Part.
However, nothing happens.
Does the Script Editor Web Part support this?
No, Script Editor web part just supports HTML/CSS/JavaScript/jQuery code, it is not available for SharePoint server web controls.
You can add the code into the page via SharePoint Designer 2013 instead of Script Editor web part.