Function to return the sum of multiples within a range
Simpler implementation

You don't need naturals at all, and offset either.
You could make the outer loop iterate over the range(start, end),
and for each value check if it can be divided by any one of nums,
and add to the sum.

msum = 0
for i in range(start, end):
    for num in nums:
        if i % num == 0:
            msum += i
            break
return msum


API

The interface of your function is not intuitive:


The range start parameter goes after the range end parameter. That's not intuitive. Consider how Python's range works:


With one arg, the first arg is the end of the range
With two args, the first arg is the start, the second is the end of the range. It's natural that way


I suggest to follow that example
The first arg is called "multiple", singular. It's not a multiple. It's a collection of numbers. So nums would be a better name, and it would make it easier for readers to understand how to use this function


Input validation

In this part:


if start == 0:
  raise ZeroDivisionError('Cannot start with the number 0')



The zero division error is inappropriate. You did not divide by zero,
and it won't make sense for callers to catch this exception and handle it.
In other languages this should be an IllegalArgumentException, in Python ValueError would be more appropriate.
The real problem is not division by zero, but inappropriate use of the function.

But then, why is start=0 inappropriate? I see nothing wrong with that.
What would be wrong is having 0 in nums.
That's what you should validate instead.

Coding style

The recommended practice is to use snake_case for function names, not CamelCase. See more coding style recommendations in PEP8.