Hi Olga,
I did a little research on this one and that functionality may not be possible with that app.
With most of our Core apps, when we encounter the {user} variable, we transform that variable into a user OID. So the query that is issued to the API looks like this:
query=(Owner = "/user/190200030006")
The Custom List with Deep Export app does not seem to perform this transform and so the query looks like this:
query=(Owner = "{user}")