Esso IS the Exxon part of ExxonMobil. Former Standard Oil Co. of NJ. Mobil was the Standard Oil Co. of NY. The two former relations remerged to make ExxonMobil. After the 1911 Standard Oil breakup - each company got the right locally to use 'Standard Oil' as a trademark.
Exxon is the brand only used in the USA, since Esso (Eastern States Standard Oil) is only legal in parts of the US that Exxon or Mobil had rights to the 'Standard Oil' trademark. Standard Oil NJ used to use three or more brands (Enco, Humble, E, Carter, etc.) across the USA with the same basic logo to circumvent the laws. A court ruled years ago that Esso sounds like 'SO' => which sounds like Standard Oil acronym, so it cannot be used where they don't have the rights. Exxon has the non-USA rights.
Esso was/is such a well known brand, they kept it alive outside of the USA.
Same reason why BP Marine stations in Ohio are still 'Sohio' or Chevron maintains a 'Standard' station with Chevron graphics in each state they still rights to. 'Caltex' in Asia/Oceania is California Standard/Texaco. The two merged years later anyway to make ChevronTexaco and later just Chevron.