I think “Ever Given” is the name of the ship and “Evergreen” is the name of the carrier company that owns it.
Yes, that’s it. “Evergreen is the name of the shipping company that leases the ship. The vessel itself is called Ever Given. Evergreen Marine, a Taiwanese cargo and freight business, has many, many ships. Ever Given is one of 20 container ships in Evergreen’s fleet and is named in the company’s “Ever” + “G-word” format. There’s also an Ever Gentle, an Ever Gleamy, an Ever Genius … you get the gist. There’s also an Ever Going...” (from https://www.abc.net.au/news/2021-03-25/ship-stuck-in-the-suez-canal-called-ever-given-plus-a-few-memes/100027880)