CharacterAdded is not firing?

According to the wiki, the backpack is deleted and replaced with a new version when your character respawns. This means that by the time your script detects the backpack, your character has already respawned so it doesn't fire the CharacterAdded event. Therefore, try changing your code to this so that it always makes sure you have the new backpack after the character gets added:

function AddTool(plr, obj)
    print(obj:GetFullName())
    if obj:IsA("Tool") or obj:IsA("HopperBin") and not plr.StarterGear[obj.Name] then
        obj:Clone().Parent  = plr.StarterGear
    end
end

game.Players.PlayerAdded:connect(function(plr)
    plr.CharacterAdded:connect(function(chr)
        print("CharacterAdded")
local bp = plr:WaitForChild("Backpack")
    bp.ChildAdded:connect(function(obj)
        AddTool(plr, obj)
    end)
        chr.ChildAdded:connect(function(obj)
            print("Child Added")
            AddTool(plr, obj)
        end)
    end)
end)

Note: I'm on an iPad so I'm not perfectly sure if it works. This is just what I observed wrong with your code.