How to deal with unknown variables or How to deal with multiple databases

If you want to preserve the interface with those methods, you should change a little your interface as:

Interface:

// The interface that all connectors should have
type Intfc interface {
    // Connect to the database, if an error occur at the moment
    // of connection, return the error
    Connect() error
    // Add returns a string, it returns an error if something occurs
    Add(string, string, string) (string, error)
    Get(string) (Object, bool)
}

MySQL:

type Mysql struct{
    conn *sql.DB // contains the connection to the DB
}

// Connect to mysql
func (f *Mysql) Connect() error {
    conn, err := sql.Open("mysql", "yourusername:yourpassword@/yourdatabase")
    if err != nil {
        return error
    }
    f.conn = conn
    return nil
}

// Add item to DB
func (f *Mysql) Add(owner string, refType string, object string) (string, error) {
    // do something
    return // a string and error
}

func (f *Mysql) Get(Uuid string) (dbinit.Object, bool) {
    // do something
    return // an object and a bool
}

Mongo:

type Mongo struct{
    session *mgo.Session
}

// Connect to mongo
func (f *Mongo) Connect() error {
    info := &mgo.DialInfo{
        // some data
    }

    session, err := mgo.DialWithInfo(info)
    if err != nil {
        return error
    }
    f.session = session
    return nil
}

// Add item to DB
func (f *Mongo) Add(owner string, refType string, object string) (string, error) {
    // do something
    return // a string and error (it could be nil at success)
}

func (f *Mongo) Get(Uuid string) (dbinit.Object, bool) {
    // do something
    return // an object and a bool
}

Main:

var selectedDb dbinit.Intfc

commandLineInput := "mysql"

if commandLineInput == "mysql" {
    selectedDb = &mysqlConnector.Mysql{}
} else if commandLineInput == "mongo" {
    selectedDb = &mongoConnector.Mongo{}
}

err := selectedDb.Connect()
if err != nil {
    panic(err)
}

// this runs everytime the API is called
api.HandlerFoobar = foobar.handlerFunction(func(params foobar.Params) middleware.Responder {
    data, err := selectedDb.Add(addStringA, addStringB, addStringC)
    if err != nil {
        // do something
    }

    return // the API response
})

But you also can remove the Connect() error method from Intfc and just use Add and Get, but you should update the packages like:

Mysql

// Connect to mysql, it could be any function name
func Connect() (*Mysql, error) {
    connection, err := sql.Open("mysql", "yourusername:yourpassword@/yourdatabase")
    if err != nil {
        return nil, error
    }
    return &Mysql{conn: connection}
}

Mongo

// Connect to mongo, it could be any function name
func Connect() (*Mongo, error) {
    info := &mgo.DialInfo{
        // some data
    }

    s, err := mgo.DialWithInfo(info)
    if err != nil {
        return nil, error
    }
    return &Mongo{session: s}
}

Main

var selectedDb dbinit.Intfc
var err error
commandLineInput := "mysql"

if commandLineInput == "mysql" {
    selectedDb, err = mysqlConnector.Connect()
} else if commandLineInput == "mongo" {
    selectedDb, err = mongoConnector.Connect()
}

if err != nil {
    panic(err)
}

I think, that interface Intfc (or better name DbIntfc) should have only methods Get and Add.

And it should exist another function - but not a part of DbIntfc, that returns DbIntfc - that connects to MySql or MongoDb. Let's look:

    type MySqlDbIntfc struct{
       db *Sql.DB
    }

    // Connect to mysql
    func NewMySqlDbIntfc() (DbIntfc,error) {
        // Please do not prefer panic in such abstract methods
        client, err := sql.Open("mysql", "yourusername:yourpassword@/yourdatabase")
        if err != nil {
            return nil, err
        }
        return &MySqlDbIntfc{client}, nil
    }

    func (mySqlDb *MySqlDbIntfc) Get(Uuid string) (dbinit.Object, error) {
      var obj dbinit.Object
      err := mySqlDb.db.QueryRow("SELECT uuid, object, deleted FROM myTable WHERE uuid=?", Uuid).Scan(&obj.Uuid, &obj.Object, &obj.Deleted)
      if err != nil {
        return dbinit.Object{}, err
      }
      return obj, nil
    }

And implemention NewMgoDbIntfc should be easy, including methods NewMgoDbIntfc.Add/Get.

And decision whether use NewMySqlDbIntfc or NewMgoDbIntfc should also be easy.

Elaborating on my comment, instead of

type Intfc interface {
    Connect() HERE_LIES_MY_PROBLEM
    Add(string, string, string) string
    Get(string) (Object, bool)
}

you can use

type Intfc interface {
    Connect() DBClient
}

and

type DBClient interface {
    Add(string, string, string) string
    Get(string) (Object, bool)
}

type MySQLClient sql.DB
type MongoClient mgo.Session

func (f Mysql) Connect() DBCLient {

    client, err = sql.Open("mysql", "yourusername:yourpassword@/yourdatabase")
    if err != nil {
        panic(err.Error())    
    }

    return MySQLClient(client)
}

func (f Mongo) Connect() DBClient {

    info := &mgo.DialInfo{
        Addrs:    []string{hosts},
        Timeout:  60 * time.Second,
        Database: database,
        Username: username,
        Password: password,
    }

    client, err := mgo.DialWithInfo(info)
    if err != nil {
        panic(err)
    }

    return MongoClient(client)
}

func (s *MySQLClient) Add(...) {
    // ...
}

func (s *MongoClient) Add(...) {
    // ...
}