When reports say certain islands are mined for phosphate, is that the same as saying they are mined for guano? I saw one reference that said Navassa Island was claimed under the Guano Island Act but that it was not guano at all but phosphate that was eventually mined there. However, many websites seem to use these terms to refer to the same thing. 
Guano is bird (or bat) excrement, and is rich in phosphates.
No. Guano contains phosphate, and is a fairly easy way to get it, but there are also mineral deposits that contain phosphate.
Guanine, the DNA base (the ‘G’ of ATGC), is named after guano, in which it can be found.
The phosphate deposits on Navassa, as well as Nauru and some other islands, are essentially petrified guano. They are ancient deposits of bird droppings that have become solidified and rocklike. There are also fresher deposits of guano that are mined, such as on the Guano Islands off the coast of Peru.
Phosphate rock or phosphorite is a sedimentary rock that contains large amounts of phosphorus, and is the other main source of phosphate fertilizer besides fresh and petrified guano.
As well as phosphorus, guano also also contains a lot of nitrogen in the form, mainly, of ammonia and uric acid. These are also important fertilizers. Indeed, although I do not know the actual figures, I would be surprised if guano does not contain a lot more nitrogen than phosphorus, and if its value does not derive as much or more from this than from its phosphorus content.
Presumably, phosphate rocks (unless they originated from fossilized guano) do not provide useful amounts of nitrogen compounds.
Don’t ask me.
Is your brother Phosphate Lad?