CanCollide is not a valid member of Model "Dog"?

CanCollide and CFrame are not properties of Models, they are properties of BaseParts.

In order to get this working, first, you need to set the PrimaryPart of the dog model. This can be done through the properties window. Then, set the CFrame of the primary part and set CanCollide of all things within the model. Also, the RunService is not being used here.

local runService = game:GetService("RunService")

local pet = game.ReplicatedStorage:WaitForChild("Dog") 
local offset = CFrame.new(2, 2, 2) -- relative location

game.Players.PlayerAdded:Connect(function(player)            player.CharacterAdded:Connect(function(character) 
local humanoidRootPart = character:WaitForChild("HumanoidRootPart") 
local humanoid = character:WaitForChild("Humanoid")

runService.RenderStepped:Connect(function()


local newPet = pet:Clone() -- New pet

for i, v in pairs(newPet:GetChildren()) do
    if v:IsA("BasePart") then
        v.CanCollide = false
    end
end
newPet:PivotTo(humanoidRootPart.CFrame * offset)

end)

end)
end)

Here's the script without RunService in use.

local runService = game:GetService("RunService")

local pet = game.ReplicatedStorage:WaitForChild("Dog") 
local offset = CFrame.new(2, 2, 2) -- relative location

game.Players.PlayerAdded:Connect(function(player)            player.CharacterAdded:Connect(function(character) 
local humanoidRootPart = character:WaitForChild("HumanoidRootPart") 
local humanoid = character:WaitForChild("Humanoid")


local newPet = pet:Clone() -- New pet

for i, v in pairs(newPet:GetChildren()) do
    if v:IsA("BasePart") then
        v.CanCollide = false
    end
end
newPet:PivotTo(humanoidRootPart.CFrame * offset)

end)
end)

Notes: If :PivotTo is giving issues, use :SetPrimaryPartCFrame instead.