Why its not submitting data in another database?

Try this one…

import wixUsers from 'wix-users';
import wixData from 'wix-data';

$w.onReady(function () {
    let user = wixUsers.currentUser;
    let userId = user.id; console.log(userId);

    wixData.query("Members/PrivateMembersData")
    .eq("_id", wixUsers.currentUser.id)
    .find()
    .then((results) => {console.log(results.items[0])
        $w('#input7').value = results.items[0].loginEmail;
        $w('#input5').value = results.items[0].firstName;
        $w('#input6').value = results.items[0].lastName;
    });

    $w('#button1').onClick(()=>{
        $w('#dataset1').setFieldValue('firstName', $w('#input5').value);
        $w('#dataset1').setFieldValue('lastName', $w('#input6').value);
        $w('#dataset1').setFieldValue('eMail', $w('#input7').value);
    });

    $w('#dataset1').onReady(()=>{
        $w('#dataset1').save()
        .then((item)=>{let fieldValue = item.fieldName; console.log(fieldValue);})
        .catch((err)=>{console.log(err);});
    });
});

EDIT: You perhaps also should consider to use → setFieldValues instead, which gives you the ability to set multiple values and save them at once…

$w("#myDataset").setFieldValues({
   'firstName', $w('#input5').value,
   'lastName', $w('#input6').value,
   'eMail', $w('#input7').value
});