But i think it will be not possible to use DSN connection with this.
If you are talking about the ODBC DSN you use to create the linked table in Access then you are correct you are limited to whatever user is specified in the ODBC Manager.
It would help to know what you plan to use the user name for?
In the meantime just an idea, but could you capture the system user in a table in Access and use that to pass on to Postgres?
How to know in postgresql which specific windows user is using database?
You cannot. All the server knows is the specific user credentials it is authenticating.
That said you can authenticate those credentials in such a way so that knowing the signed on user you would also know who they are in any environment that uses the same authentication source - and if that source supplies their Windows identity you are golden. The specific setups involved here are outside my experience, though.