Password generator in python

Refactoring in steps

Enumeration class

Types is too generic name for the enum representing char types. Renamed to CharTypes.
Instead CAP, SMA, DIG, SPE as enumeration members are better replaced with a more common/familiar and comprehensive abbreviations/associations:
UPPER, LOWER, DIGIT and SPECIAL.
Since string.ascii_uppercase and other string.* are essentially just string constants - they can be easily set as enumeration values:

class CharTypes(Enum):
    UPPER = string.ascii_uppercase     # Capital
    LOWER = string.ascii_lowercase     # Small
    DIGIT = string.digits              # Digits
    SPECIAL = '!()-.?[]_`~;:@#$%^&*='  # Special

thus, making all intermediate re-mappings like type_chars and types (in password_generator function) redundant and unnecessary.

password_generator function

The function signature is slightly changed in arguments names to conform with CharTypes members:

def password_generator(min_length=6, max_length=20, upper=1, lower=1, digits=1, special=1)

types mapping is eliminated as redundant.

char counts passed as arguments are gathered and summed at once:

char_counts = (upper, lower, digits, special)
num_chars = sum(char_counts)

Avoid overwriting/assigning to function argument like min_length = max(num_chars, min_length) as min_length might be potentially referenced as "original" argument value (and relied on) in other places in the function's body.
A safer way is assigning it to a separate variable:

min_len = max(num_chars, min_length)

length variable is renamed to target_length (to emphasize the final size).

char_list is renamed to char_types as it's aimed to accumulate CharTypes enum members

Two for loops which performed char_list.extend and char_list.append are efficiently replaced with 2 generators which further joined/merged by itertools.chain function:

char_types = list(chain(*([c_type] * num for c_type, num in zip(CharTypes, char_counts)),
                        (choice(char_types_enums) for _ in range(target_length - num_chars))))

Furthermore, itertools.chain is smart enough to skip empty generators (if let's say there's no remaining values to fill).

The last for loop (accumulating password from random chars) is simply replaced with str.join call on generator expression:

password = ''.join(choice(char_type.value) for char_type in char_types)

The whole crucial functionality is now shortened to the following:

import string
from enum import Enum
from random import randint, shuffle, choice
from itertools import chain


class PasswordError(Exception):
    pass


class CharTypes(Enum):
    UPPER = string.ascii_uppercase     # Capital
    LOWER = string.ascii_lowercase     # Small
    DIGIT = string.digits              # Digits
    SPECIAL = '!()-.?[]_`~;:@#$%^&*='  # Special


def password_generator(min_length=6, max_length=20, upper=1, lower=1, digits=1, special=1):
    char_counts = (upper, lower, digits, special)
    num_chars = sum(char_counts)   # Number of mandatory characters
    min_len = max(num_chars, min_length)   # In case 'num_chars' is greater

    # If number of characters required for each possible char type
    # is greater than maximum possible length
    if min_len > max_length:
        raise PasswordError(f'No password with the given criteria')

    target_length = randint(min_len, max_length)
    char_types_enums = list(CharTypes)  # get list of enums to pass `random.choice` call

    # List of char "types" comprised of: mandatory requirements + remaining values to fill
    char_types = list(chain(*([c_type] * num for c_type, num in zip(CharTypes, char_counts)),
                            (choice(char_types_enums) for _ in range(target_length - num_chars))))
    shuffle(char_types)

    password = ''.join(choice(char_type.value) for char_type in char_types)
    return password


if __name__ == '__main__':
    ....

Sample usage:

Minimum number of characters required: >? 10
Maximum number of characters possible: >? 30
Number of capital letters required: >? 5
Number of small letters required: >? 4
Number of digits required: >? 6
Number of special characters required: >? 5
Number of passwords required: >? 4
--------------------------------------------------

   Here are your passwords: 

32S%km3A^v04h9pwR-T7O;=0O
mh8a:38Q-pGS3PtGs)e0P1g)$(#0U1
z@a0r;b7v.~K!8S@R343J7L
Mie:8Ec0C=3Cz93HPHDFm_84#;6@