When joining or buying skin amount doesn't update?

General Practice

Use :WaitForChild() to determine if a part exists before using / changing it (you did do this, but to check warnings, it's best to continue this throughout the script)

Use :GetService() to retrieve the Players and ReplicatedStorage Services

Issues

Do not use :FindFirstChild() on a Table, either iterate through it or call the index value directly

You cannot perform math on a string, so you need to change it to a number using tonumber() before doing arithmetic

Revised Local Script V1

local player = game:GetService("Players").LocalPlayer
local ownedSkins = player:WaitForChild("PlayerSkins")
local chosenKnife = player:WaitForChild("SelectedKnifeSkin")
local chosenGun = player:WaitForChild("SelectedGunSkin")
local skins = game:GetService("ReplicatedStorage"):WaitForChild("Skins")
local template = game:GetService("ReplicatedStorage"):WaitForChild("SkinTemplate")
local scroll = script.Parent:WaitForChild("Frame"):WaitForChild("ScrollingFrame")
local invSkins = nil

ownedSkins.ChildAdded:Connect(function(child)
    if scroll:FindFirstChild(child.Name) == nil then
	    local rarity = child:WaitForChild("Rarity")
	    local nTemplate = template:Clone()
	    local label = nTemplate:WaitForChild("TextLabel")
	    
	    nTemplate.Parent = player:WaitForChild("PlayerGui"):WaitForChild("Menu"):WaitForChild("Inventory"):WaitForChild("Frame"):WaitForChild("ScrollingFrame")
	    nTemplate.Image = ("rbxassetid://"..(child:WaitForChild("Icon").Value))
	    nTemplate.Name = child.Name
	    label.Text = child.Name
	    
	    if rarity.Value == "Common" then
			label.TextColor3 = Color3.fromRGB(140, 139, 144)
	    elseif rarity.Value == "Uncommon" then
		    label.TextColor3 = Color3.fromRGB(75, 88, 212)
	    elseif rarity.Value == "Legendary" then
		    label.TextColor3 = Color3.fromRGB(255, 198, 83)
	    end
    elseif scroll:FindFirstChild(child.Name) ~= nil then
	    local skinInv = scroll:FindFirstChild(child.Name)
	    local label = skinInv:WaitForChild("Amount"):WaitForChild("TextLabel")
	    label.Text = tonumber(label.Text) + 1
    end
    invSkins = scroll:GetChildren()
end)

Explanation #1

The main issue with your code is that multiple frames are being created, which can only occur if the "child.Name" is not the same as any of the children of the ScrollingFrame. However, since this code doesn't show how the values are added to the player.PlayerSkins folder, then I can't extrapolate where the difference occurs just reading your code. However, I will instead give you a revision below that will help you determine what the change is.

Example Code

Below, the output should print out the child's name and the names of the children of the Scrolling Frame, as well as a statement denoting if the name of the child and the scroll item match (it will print nil if they don't), which is returned by string.match()

    print("The child's name is "..child.Name)
    for a, b in pairs(scroll:GetChildren()) do
		print("The scroll item's name is "..b.Name)
		print("Match Result: "..string.match(b.Name, child.Name, 1))
	end

Revised Local Script V2

local player = game:GetService("Players").LocalPlayer
local ownedSkins = player:WaitForChild("PlayerSkins")
local chosenKnife = player:WaitForChild("SelectedKnifeSkin")
local chosenGun = player:WaitForChild("SelectedGunSkin")
local skins = game:GetService("ReplicatedStorage"):WaitForChild("Skins")
local template = game:GetService("ReplicatedStorage"):WaitForChild("SkinTemplate")
local scroll = script.Parent:WaitForChild("Frame"):WaitForChild("ScrollingFrame")
local invSkins = nil

ownedSkins.ChildAdded:Connect(function(child)
    print("The child's name is "..child.Name)
    for a, b in pairs(scroll:GetChildren()) do
		print("The scroll item's name is "..b.Name)
		print("Match Result: "..string.match(b.Name, child.Name, 1))
	end
	
    if scroll:FindFirstChild(child.Name) == nil then
	    local rarity = child:WaitForChild("Rarity")
	    local nTemplate = template:Clone()
	    local label = nTemplate:WaitForChild("TextLabel")
	    
	    nTemplate.Parent = player:WaitForChild("PlayerGui"):WaitForChild("Menu"):WaitForChild("Inventory"):WaitForChild("Frame"):WaitForChild("ScrollingFrame")
	    nTemplate.Image = ("rbxassetid://"..(child:WaitForChild("Icon").Value))
	    nTemplate.Name = child.Name
	    label.Text = child.Name
	    
	    if rarity.Value == "Common" then
			label.TextColor3 = Color3.fromRGB(140, 139, 144)
	    elseif rarity.Value == "Uncommon" then
		    label.TextColor3 = Color3.fromRGB(75, 88, 212)
	    elseif rarity.Value == "Legendary" then
		    label.TextColor3 = Color3.fromRGB(255, 198, 83)
	    end
    elseif scroll:FindFirstChild(child.Name) ~= nil then
	    local skinInv = scroll:FindFirstChild(child.Name)
	    local label = skinInv:WaitForChild("Amount"):WaitForChild("TextLabel")
	    label.Text = tonumber(label.Text) + 1
    end
    invSkins = scroll:GetChildren()
end)