As Patrick said in the comments, your main problem is that you define your own __init__ functions when using @dataclass. If you delete it and slightly restructure your code, it should work as expected:

from dataclasses import dataclass
from typing import List

@dataclass
class Package:
    _address: List[str]

@dataclass
class Destination:
    _start: str
    _end: str
    _distance: float

@dataclass
class DestinationContainer:
    _package: Package
    _destinations: List[Destination]

    def addPkg(self, param):
        # sure this shouldn't be "self._package.append(param)"? 
        self._package = param


# works
dc = DestinationContainer(
    Package(['some address']),
    [Destination('s', 'e', 1.0)]
)
# also works
dc.addPkg(Package(['some other address']))

There is no Array datatype, but you can specify the type of my_array to be typing.List:

from dataclasses import dataclass
from typing import List

@dataclass
class Test:
    my_array: List[ChildType]

And from Python 3.9 onwards, you can conveniently just use list:

from dataclasses import dataclass

@dataclass
class Test:
    my_array: list[ChildType]

A common and efficient approach to keeping track of all instances of a class is to store each instance in a class variable of a dict by a certain key, and use a class method to retrieve an instance by the key. In your case, use the id attribute as the key:

from typing import ClassVar
from dataclasses import dataclass

@dataclass
class Feld:
    _id_to_instance: ClassVar[dict] = {}
    id: str
    value: str

    def __post_init__(self):
        self._id_to_instance[self.id] = self

    @classmethod
    def get_instance_by_id(cls, id):
        return cls._id_to_instance[id]

so that:

for id, value in ('id1', 'value1'), ('id2', 'value2'):
    Feld(id, value)

print(Feld.get_instance_by_id('id2'))

outputs:

Feld(id='id2', value='value2')

Demo: https://ideone.com/KgQH5y

The answer depends on whether or not you have access to an object of the class.

Just using the class

If you only have access to the class, then you can use dataclasses.fields(C) which returns a list of field objects (each of which has a .name property):

[field.name for field in dataclasses.fields(C)]

From an existing object

If you have a constructed object of the class, then you have two additional options:

1. Use dataclasses.fields on the object:

[field.name for field in dataclasses.fields(obj)]

2. Use dataclasses.asdict(obj) (as pointed out by this answer) which returns a dictionary from field name to field value. It sounds like you are only interested in the .keys() of the dictionary:

dataclasses.asdict(obj).keys()       # gives a dict_keys object
list(dataclasses.asdict(obj).keys()) # gives a list
list(dataclasses.asdict(obj))        # same 

Full example

Here are all of the options using your example:

from dataclasses import dataclass, fields, asdict


@dataclass
class C:
    x: int
    y: int
    z: int
    t: int

# from the class
print([field.name for field in fields(C)])

# using an object
obj = C(1, 2, 3, 4)

print([field.name for field in fields(obj)])
print(asdict(obj).keys())
print(list(asdict(obj).keys()))
print(list(asdict(obj)))

Output:

['x', 'y', 'z', 't']
['x', 'y', 'z', 't']
dict_keys(['x', 'y', 'z', 't'])
['x', 'y', 'z', 't']
['x', 'y', 'z', 't']