mypy is really handy and catches a lot of bugs, but when I write "scientific" applications, I often end up doing:

def my_func(number: Union[float, int]):
    # Do something

number is either a float or int, depending on the user's input. Is there an official way to do that?

Use float only, as int is implied in that type:

def my_func(number: float):

PEP 484 Type Hints specifically states that:

Rather than requiring that users write import numbers and then use numbers.Float etc., this PEP proposes a straightforward shortcut that is almost as effective: when an argument is annotated as having type float, an argument of type int is acceptable; similar, for an argument annotated as having type complex, arguments of type float or int are acceptable.

(Bold emphasis mine).

Ideally you would still use numbers.Real:

from numbers import Real

def my_func(number: Real):

as that would accept fractions.Fraction() and decimal.Decimal() objects as well; the number pyramid is broader than just integers and floating point values.

However, these are not currently working when using mypy to do your type checking, see Mypy #3186.