This still seems to happen in 2022.1

It will recur on 2019.3.10.

Problems when dragging a group of toggles after selecting multiple toggles in Hierarchy.
If you add a toggle group to one toggle, and copy it, there is no problem.

I can confirm this is fixed in 2019.1.10f1

2019.1.10f1 fixed it.

Update (for anyone looking for a more lightweight workaround)

Ultimately, all you need to avoid this issue is for "toggleGroup.allowSwitchOff" to be true before your "menu" is "disabled," which triggers the bug. This is a friggin' hassle to do, but since I already have delegates which are called when enabling and disabling the objects that contain my ToggleGroups, I can change allowSwitchOff to be false only while the ToggleGroup is active.

This is dependent on my own "MenuController" class and its "onMenuEnabled/onMenuDisabled" delegates, so you'll have to adjust this to fit your codebase as needed, but conceptually, it's a drop-in solution that doesn't add a lot of overhead at runtime. I also set this script to run early in the project's Script Execution Order, just in case.

(My actual version of this code populates the needed variables at *build-time* using IProcessSceneWithReport, so that I can just drop this on a object with a ToggleGroup without having to configure it while also avoiding any overhead (however tiny) in Start or Awake.)

.........

public class ToggleGroupBugFixer : MonoBehaviour {

// This is to fix a dumb bug in Unity 2019.1.

[SerializeField] private MenuController menu;
[SerializeField] private ToggleGroup toggleGroup;

private void Awake(){
menu.onMenuEnabled += menuEnabledHander;
menu.onMenuDisabled += menuDisabledHandler;
}
private void OnDestroy(){
menu.onMenuEnabled -= menuEnabledHander;
menu.onMenuDisabled -= menuDisabledHandler;
}

private void menuEnabledHander(){
toggleGroup.allowSwitchOff = false;
}
private void menuDisabledHandler(){
toggleGroup.allowSwitchOff = true;
}

}

CarlosFi's "failsafe" workaround basically works for me (thank you!!), but since I have other listeners checking onValueChanged, I end up getting stack overflows when the failsafe kicks in, so not ideal (ultimately I'll be wasting a few hours implementing and testing this or another workaround... ugh).

This is a preposterous issue. Apart from the fact that "upgrading" to an alpha or beta version not being a serious solution, some of us are stuck on slightly older versions for 3rd-party SDK reasons which we have no control over, so this is a perfect example of why Unity needs to admit defeat and go back to issuing minor "f" releases to fix tiny but utterly game-breaking bugs like this.

Even with a workaround in place, I don't think I can ship with a bug this silly.

Updating to 2019.1.9f1 did not fix the issue. Hopefully next update will fix it.

"Fixed in 2019.1" which version exactly?
In 2019.1.9f1 there is no word in changelog about this issue.

Sorry... my workaround code could be simplified even more removing some unecesary lines:

using System.Collections;
using System.Collections.Generic;
using UnityEngine;
using UnityEngine.UI;

public class ToggleGroupHelper : MonoBehaviour
{
private Toggle rememberMe;

List<Toggle> source_list;

private System.Reflection.FieldInfo _toggleListMember;
ToggleGroup m_Group;

void Start()
{
m_Group = this.GetComponent<ToggleGroup>();

if(m_Group.allowSwitchOff)
{
return;
}
m_Group.allowSwitchOff = true;

if (_toggleListMember == null)
{
_toggleListMember = typeof(ToggleGroup).GetField("m_Toggles", System.Reflection.BindingFlags.Instance | System.Reflection.BindingFlags.NonPublic);
if (_toggleListMember == null)
throw new System.Exception("UnityEngine.UI.ToggleGroup Has become public, so this code must be changed!");
}
source_list = _toggleListMember.GetValue(m_Group) as List<Toggle>;
if(source_list != null)
{
foreach(Toggle tt in source_list)
{
tt.onValueChanged.AddListener((bool on) => { ToggleChanged(tt, on); });
}
}
}

private void ToggleChanged(Toggle tt, bool on)
{
if (!on)
{
if(!m_Group.AnyTogglesOn())
{
tt.isOn = true;
}
}
}
}

ELADLEB4 workaround did not work well for me, so I coded another helper class that fixes the bug until 2019.3 version is available.
Just create a class called ToggleGroupHelper and attach this script as a component to the same object where the ToggleGroup component is:

[code=CSharp]
using System.Collections;
using System.Collections.Generic;
using UnityEngine;
using UnityEngine.UI;

public class ToggleGroupHelper : MonoBehaviour
{
List<Toggle> source_list;
List<Toggle> toggle_list = new List<Toggle>();

private System.Reflection.FieldInfo _toggleListMember;
ToggleGroup m_Group;

void Start()
{
m_Group = this.GetComponent<ToggleGroup>();

if(m_Group.allowSwitchOff)
{
return;
}
m_Group.allowSwitchOff = true;

if (_toggleListMember == null)
{
_toggleListMember = typeof(ToggleGroup).GetField("m_Toggles", System.Reflection.BindingFlags.Instance | System.Reflection.BindingFlags.NonPublic);
if (_toggleListMember == null)
throw new System.Exception("UnityEngine.UI.ToggleGroup Has become public, so this code must be changed!");
}
source_list = _toggleListMember.GetValue(m_Group) as List<Toggle>;
if(source_list != null)
{
foreach(Toggle tt in source_list)
{
toggle_list.Add(tt);
tt.onValueChanged.AddListener((bool on) => { ToggleChanged(tt, on); });
}
}
}

private void ToggleChanged(Toggle tt, bool on)
{
if (!on)
{
if(!m_Group.AnyTogglesOn())
{
tt.isOn = true;
}
}
}
}[/code]