Why this if clause only work for the first sentences??

This don't works because of coding logic(don't worry, i'll explain), so, basically, that variable definition is just a logic for code, waiting for the first true:

If you do

local var = "Text" or "Other text"

var will aways be "Text"

thats because of logic, basically, your code is trying to get a true on that, and because its a string, its aways true.

local var = (var1 == 1) and "text" or "other text"

^^ this is also coding logic, basically, it will set the variable to "text" if var1 is equals to 1, if its not, it will set to "other text"

I know this seems complicated, but its not easy to explain it.

Now into your code, that variable Undeletable is aways "Golden Dominus", what you can do is defining that strings inside a table, and then checking if equals to any of the values.

Here's how to do it:

local Undeletable = { -- Start declaration of table
"Golden Dominus", "Aqua Dragon", "Dominus Aura", "Shiny Dog", "Shiny Cat", "Shiny Chicken", "Shiny Turtle"
} -- end declaration of table (( the table can be all on the same line ))

Now, your variable is a table, so, how to check the values?

We can use a for ... do loop:

v.MouseButton1Click:Connect(function()
    local cantdelete = false -- here, we create a variable on the MouseButton1Click scope
    for _,v in pairs(Undeletable) do
        if PetNames == v then
            cantdelete = true  -- if the object can't be deleted, we change that variable to true
            break -- and break the loop, so it wont run any more, as its not needed
        end
    end
    -- now we can simply check if that variable we defined is true or false
    if cantdelete then -- if object cant be deleted
        delete.Visible = false
    else
        delete.Visible = true
    end

    if PetNames == Undel then -- variable undel isn't defined
        shiny.Visible = false
    else
        shiny.Visible = true
    end
end)

Useful links:

DevAPI/Scopes

DevAPI/Loops#for-do

DevAPI/Tables

DevAPI/Conditional Structures

if this helps, please upvote and mark this answer as accepted