How to set up a user who doesn't have `admin` database access but use it as authentication database?
Not need. You just give user needed rights to the wanted non-admin database. User can use admin database as authentication database even user doesn't have read/write access to admin database.

use products
db.grantRolesToUser("productsUser",[ "readWrite" ])


grantRolesToUser documentation.

UPDATE

Let's create mongodb instance with --auth. Login as admin, create user what can readWrite ONLY test -db, authenticate with that user, check can use list admin database collections and what this user can do with test database.

#&gt; mlaunch init --single --auth
launching: mongod on port 27017
Username "user", password "password"
#&gt; mongo -u user -p password admin
MongoDB shell version v3.4.6
connecting to: mongodb://127.0.0.1:27017/admin
MongoDB server version: 3.4.6
Mongo&gt; db.createUser({user:"test", pwd:"testpwd", roles:[{role:"readWrite", db:"test"}]})
Successfully added user: {
    "user" : "test",
    "roles" : [
        {
            "role" : "readWrite",
            "db" : "test"
        }
    ]
}
Mongo&gt; db.auth("test","testpwd")
1
Mongo&gt; db
admin
Mongo&gt; show collections
2017-09-11T19:06:22.001+0300 E QUERY    [thread1] Error: listCollections failed: {
    "ok" : 0,
    "errmsg" : "not authorized on admin to execute command { listCollections: 1.0, filter: {} }",
    "code" : 13,
    "codeName" : "Unauthorized"
} :
_getErrorWithCode@src/mongo/shell/utils.js:25:13
DB.prototype._getCollectionInfosCommand@src/mongo/shell/db.js:807:1
DB.prototype.getCollectionInfos@src/mongo/shell/db.js:819:19
DB.prototype.getCollectionNames@src/mongo/shell/db.js:830:16
shellHelper.show@src/mongo/shell/utils.js:762:9
shellHelper@src/mongo/shell/utils.js:659:15
@(shellhelp2):1:1
Mongo&gt; use test
switched to db test
Mongo&gt; db.coll.insert({})
WriteResult({ "nInserted" : 1 })
Mongo&gt; show collections
coll
Mongo&gt;  


And same from outside:

#&gt; mongo -u test -p testpwd --authenticationDatabase test
MongoDB shell version v3.4.6
connecting to: mongodb://127.0.0.1:27017
MongoDB server version: 3.4.6
2017-09-11T19:17:48.614+0300 E QUERY    [thread1] Error: Authentication failed. :
DB.prototype._authOrThrow@src/mongo/shell/db.js:1461:20
@(auth):6:1
@(auth):1:2
exception: login failed
#&gt; mongo -u test -p testpwd --authenticationDatabase admin test
MongoDB shell version v3.4.6
connecting to: mongodb://127.0.0.1:27017/test
MongoDB server version: 3.4.6
Mongo&gt;