Creating a CSV is as simple as writing a string with each value separated by a designated character, often a comma as the name implies. You don't really need the csv library for that.

Your Signup method would then look as follows:

def Signup(j,k):
    print("Welcome to sign up!\n Please enter your Username")
    j = input("")
    if j not in Brugere:
        print("Hello, " + j + ", please enter a password:")
        k = input("")
        line1 = [j,k]
        # Write a string with a comma between the values   
        Brugere.write("{}, {}\n".format(j, k))
        print("Great! Now you can sign in")
        Signin(Brugernavn, Kodeord)
    else:
        print("Username already taken! try again")
        Signup(Brugernavn, Kodeord)

With that being said, using CSVs to store operational data in a an application does not make all that much sense to me. Please see below for a modified version of signup using SQLite, which is much more robust and still simple to use:

import sqlite3
con = sqlite3.connect('brugere.db')
cur = con.cursor()

# Create a user table with bruger and kodeord
# I added an extra field for when the user was created as well 
# which defaults to the date and time the user was created 
cur.execute("""create table if not exists brugere
               (brugernavn text, 
                kodeord text,
                dato_oprettet TIMESTAMP  DEFAULT CURRENT_TIMESTAMP)""")

con.close()


def signup():
    
    con = sqlite3.connect('brugere.db')
    cur = con.cursor()
    
    with con:
        print("Please enter your Username")
        bruger = input()
        cur.execute("select brugernavn from brugere where brugernavn =?", [bruger])
        
        bruger_eksisterer = cur.fetchone()
        
        if bruger_eksisterer:
            print("Username already taken! Try again")
            # Signup(Brugernavn, Kodeord)
        else:    
            print("Hello, " + bruger + ", please enter a password:")
            kodeord = input()
            
            try:
                cur.execute("insert into brugere (brugernavn, kodeord) values (?, ?)", (bruger, kodeord))
                print("Great! Now you can sign in")
            except sqlite3.Error as Err:
                print("Couldn't add user.\nError: {}".format(Err))         
                # Signin(Brugernavn, Kodeord)        
    
    con.close()

signup()    

Modifying the signin should be straighforward, but I'll leave that part up to you.

TL;DR:

I suspect the simplest solution is:

...
    if j not in brugere:
        ...
        writer.writerow(line1)
        brugere.flush()  # Flush pending changes to file.
        ...
...

However, I've noticed a few things worthy of consideration:

• Python help. You can get help on Python things within an interactive Python session:

python

> import csv
> help(csv)

• CSV for this sort of usecase isn't a good idea. If it's production grade use a proper DB, if you're just playing around locally to get a feel for Python use a dict() and store as JSON. I.e.:

import json

username = "myuser"
password = "password"
filename = "data.json"

# Reads data. Requires file `data.json` to exist, with content `{}`.
with open(filename, "r") as data_stream:
    data = json.load(data_stream)

if username in data:
    print(f"User {username} exists.")
else:
    data[username] = password  # You'd usually hash and salt a password. It's a separate topic.

# Writes data to disk.
with open(filename, "w") as data_stream:
    data_stream.write(json.dumps(data, indent=2))  # Makes it look pretty on disk.

• Note that open() returns a stream wrapped in class io.TextIOWrapper (see python -c "import io; help(io.TextIOWrapper)"). It's basically an interface to get the data you want, not the data itself. Try open("test.txt").readlines() to actually read data from stream, for example. Note that streams are consumables.

• When you call in on the stream Brugere you're calling the __contains__() dunder method on class io.TextIOWrapper which isn't implemented. But Python being Python it returns False (it's not meaningful).

• The CSV interfaces are pretty straightforward:

import csv

filename = "test.txt"

f = open(filename, "w")
writer = csv.writer(f)
writer.write(["myuser", "password"])

# Flushes lines to file.
f.flush()
...

f.close()