I just noticied that Update manager as no update for ESX 5. Is there a different source to add to get those updates.
Where are you looking at? Patch Repository? You should be able to see them under Host>UpdateManager>Patches.
Not sure what you are looking for exactly,, From the Screenshot which you have attached you can see embeddeESX 5.0.0 under product section
Which will have patched for esx 5.0
Everything looks good to me.
In VUM EmbeddedESX and ESXi mean the same. I think this is what may be confusing you.
