Pickling and copying exceptions doesn't preserve non-__dict__ attributes

GitHub fields:

assignee = None
closed_at = None
created_at = <Date 2017-04-06.05:59:35.082>
labels = ['interpreter-core', 'type-bug', '3.11']
title = "Pickling and copying exceptions doesn't preserve non-__dict__ attributes"
updated_at = <Date 2021-06-27.21:37:57.332>
user = 'https://github.com/serhiy-storchaka'

bugs.python.org fields:

activity = <Date 2021-06-27.21:37:57.332>
actor = 'iritkatriel'
assignee = 'none'
closed = False
closed_date = None
closer = None
components = ['Interpreter Core']
creation = <Date 2017-04-06.05:59:35.082>
creator = 'serhiy.storchaka'
dependencies = ['29998']
files = []
hgrepos = []
issue_num = 30005
keywords = []
message_count = 2.0
messages = ['291212', '396602']
nosy_count = 3.0
nosy_names = ['alexandre.vassalotti', 'serhiy.storchaka', 'iritkatriel']
pr_nums = []
priority = 'normal'
resolution = None
stage = 'needs patch'
status = 'open'
superseder = None
type = 'behavior'
url = 'https://bugs.python.org/issue30005'
versions = ['Python 3.11']