When the except block handles an exception, the factorial function does not explicitly return a value, so the function returns None. Then, the None is passed to the print function. It is as though the code is written like this:
result = factorial("xyz")  # result is None
print(result)              # this is where "None" is printed
I think the best way to solve this is to separate parsing the number from the factorial function:
def factorial(number):
    """ This function will calculate the factorial (N!) of a number """
    
    output = "null"
    if number > 0:
        return number * factorial(number - 1)
    elif number == 0:
        return 1
    else:
        return output
try:
    # first, parse the input
    number = int("xyz")
    # Now that `number` is guaranteed to be an integer, pass it
    # to the factorial function.
    print(factorial(number))
except:
    # If the input was not an integer, this is where the
    # exception is handled. The print function was not called
    # in the try block, so there is no extra "None" in the output
    print("Error, please ensure you are inputting a integer.")
This works, but the factorial function now sometimes returns a number and sometimes returns the string "null". It would be better if it either returned a number, or it threw an exception. We can raise a ValueError if the user provides a negative number:
def factorial(number):
    """ This function will calculate the factorial (N!) of a number """
    
    if number > 0:
        return number * factorial(number - 1)
    elif number == 0:
        return 1
    else:
        raise ValueError("Number must not be negative")
def parse_and_compute_factorial(input):
    try:
        # first, parse the input
        number = int(input)
    except ValueError:
        print("Error, please ensure you are inputting a integer.")
        return
    try:
        print(factorial(number))
    except ValueError as err:
        print(err)
parse_and_compute_factorial("xyz") # prints "Error, please ensure you are inputting a integer"
parse_and_compute_factorial("-5")  # prints "Number must not be negative"
parse_and_compute_factorial("5")   # prints "120"