There's a missing link in your chain.
First of all, the Coromandel Coast is named for the Tamil word "Cholamandalam", which means "land of the Cholas", an ancient dynasty that ruled the area once upon a time. Since Europeans had a difficult time pronouncing the Tamil word, the name was basically corrupted to "Coromandel".
Second, and here's your missing link, there was a British Navy vessel named HMS Coromandel (although it was originally named HMS Malabar). In 1820, under Captain James Downie, it was sent to New Zealand to get local kauri tree logs that were used by the Royal Navy to build ships, since the wood is both long and strong, and excellent for masts and spars.
Finally, Captain Downie, being the typical European traveller that he was, named the location that he anchored in Coromandel Harbour. As the European shipping route expanded and the tree harvesting expanded up the peninsula, the town, the bay, and eventually the entire peninsula was named Coromandel. For the record, the indigenous Maori name for the peninsula is Te Tara-o-te-Ika-a-Maui.
Now, if you want to take it one step further, the Coromandel Valley, South Australia, is also named for a ship, though not the same ship, also called Coromandel. This one was built in Quebec and brought New Zealand Company settlers to the area in 1836.
https://en.wikipedia.org/wiki/Coromandel,_New_Zealand
https://en.wikipedia.org/wiki/HMS_Malabar_(1804)
https://en.wikipedia.org/wiki/Coromandel_Coast
https://en.wikipedia.org/wiki/Coromandel_Valley,_South_Australia
