How to create a static instance of a class inside the same class?

Usually doing that is frowned upon, as it creates circular references, problematic especially when you generate the class dynamically inside functions.

If that is not your case, you can create it like so:

@dataclass
class TestClass:
    attribute1: str
    attribute2: int


TestClass.STATIC_INSTANCE1 = TestClass("instance1", 1)
TestClass.STATIC_INSTANCE2 = TestClass("instance2", 1)

I suggest you to avoid doing it altogether, and just defining STATIC_INSTANCE1 outside of the class, but it will require additional from...imports on the user side.

This is known as the singleton design pattern. It's most widely-used in Java, and as far as I know the textbook implementation is to use a method to initialize the static instance when necessary and no sooner, and to obtain that static instance:

@dataclass
class TestClass:
    attribute1: str
    attribute2: int

    _STATIC_INSTANCE1 = None
    _STATIC_INSTANCE2 = None

    @staticmethod
    def get_static_instance1():
        if TestClass._STATIC_INSTANCE1 is None:
            TestClass._STATIC_INSTANCE1 = TestClass("instance1", 1)
        return TestClass._STATIC_INSTANCE1

    # ditto for static instance 2

However, since, unlike Java, python allows free-floating objects (not everything needs to be a class), please consider whether these static instances actually need to be bound to the TestClass itself, or whether they can float in the same file:

@dataclass
class TestClass:
    attribute1: str
    attribute2: int
    ...
    pass

STATIC_INSTANCE1 = TestClass("instance1", 1)
STATIC_INSTANCE2 = TestClass("instance2", 1)

# import STATIC_INSTANCE1 and STATIC_INSTANCE2 from the same file as TestClass, 
# but separately

This latter approach is preferred in Python when possible.