Agates mostly occur as nodules in volcanic rocks or waters containing Silica in solution which is possibly derived from the decomposition of Silicates in the lava. The Siliceous matter is deposited in regular layers resulting in the banded or striped formation. It is very unlikely that you would ever find an exact duplicate of any type of Agate, Brazilian or other.
Coral is built by secreting calcium carbonate skeletons. Hard coral is fossilized over millions of years. It can be almost any color. The stony Corals are often found in Coral reefs, large structures generally found in tropical waters. Reefs are extremely diverse consisting of over 4,000 species of fish, cnidarians, mollusks, crustaceans, and other animals.