Hello everyone,
The issue's fixed now. The user was admin on the server, and in SSAS config there is a property <BuiltinAdminsAreServerAdmins>1</BuiltinAdminsAreServerAdmins>.
As a result, even if the user is not shown as an admin in SSAS, he was, because of server admin and this property.
So, either changing property to 0 or removing user from server admins solved the issue.
Thank you everyone who tried to help.