issue is with foreign key:

The Place statement conflicted using the FOREIGN KEY constraint "FK_uzytkownik_Logowanie". The conflict happened in database "Restauracja", table "dbo.Logowanie", column 'LoginID'.
The statement continues to be ended.

I take a look using breakpoints, and primary type in Logowanie table was added when breakpoints (running application) was after

baza.SubmitChanges();

Primary key of LoginID in logowanie table is added instantly throughout SubmitChanges.

How you can copy worth of LoginID from logowanie table to LoginID in uztkownik table? I add foreign key value here, but here LoginID has not value yet.

Logowanie newlog = new Logowanie()
{
   Login = model.LoginModel.Użytkownik,
   Haslo = model.LoginModel.Hasło,
   konto = model.LoginModel.Konto
};

uzytkownik user = new uzytkownik()
{
   imie = model.uzytkownikModle.imie,
   nazwisko = model.uzytkownikModle.nazwisko,
   pesel = model.uzytkownikModle.pesel,
   nip = model.uzytkownikModle.nip,
   telefon = model.uzytkownikModle.telefon,
   adres_zamieszkania = model.uzytkownikModle.adres_zamieszkania,
   email = model.uzytkownikModle.email,
   LoginID = newlog.LoginID //<<<----------------
};

baza.Logowanies.InsertOnSubmit(newlog);
baza.uzytkowniks.InsertOnSubmit(user);

baza.SubmitChanges();

uzytkownik must have a house known as Logowanie, which you'll set towards the new instance you produced but did not Submit yet:

user.Logowanie = newlog;

Becasue it is an overseas key relationship, your Logowanie class should have a property for IQueryable<uzytkownik> that's accustomed to relate the objects correctly inside a one-to-many manner. Consequently, there's no more a necessity to assign the foreign key by hand, it is possible with the object heirarchy:

Logowanie newlog = new Logowanie()
{
   Login = model.LoginModel.Użytkownik,
   Haslo = model.LoginModel.Hasło,
   konto = model.LoginModel.Konto
};

uzytkownik user = new uzytkownik()
{
   imie = model.uzytkownikModle.imie,
   nazwisko = model.uzytkownikModle.nazwisko,
   pesel = model.uzytkownikModle.pesel,
   nip = model.uzytkownikModle.nip,
   telefon = model.uzytkownikModle.telefon,
   adres_zamieszkania = model.uzytkownikModle.adres_zamieszkania,
   email = model.uzytkownikModle.email,
   // Remove this line entirely   LoginID = newlog.LoginID //<<<----------------
};

// Add the child object (user) to your Logowanie object (newlog)
newlog.uzytkowniks.Add(user)  // This may just be uzytkownik. Pluralization in LINQ is weird

// It is not necessary to queue the user object for insertion
// the newlog object will do that for you
baza.Logowanies.InsertOnSubmit(newlog);
// baza.uzytkowniks.InsertOnSubmit(user);

// SubmitChanges at this point will insert all of the children
// of newlog and assign the IDs properly
baza.SubmitChanges();

// at this point you should be able to get the newlog.LoginID
// and the user.UserID (assuming this is what you called it)

Initially when i first went across this it appeared just a little backward in my experience, but when you break the associations lower to their class representations it truly begins to create sense. It is also fantastic it does all the relationship projects for you personally.