How did Madara really obtain his Eternal Mangekyou Sharingan?
Based on the data that is provided by the story, we don't have enough information to make a deduction. The only witnesses to the event were Madara and Izuna.

While this means that Madara's account has greater credibillity than anyone else, Madara has a reputation for deceiving people, such as Obito, with complex webs of lies and staged events. As a result, we cannot consider him a reliable information source.

However, based on information provided in the Naruto Data Book (Number 4, page 37) on Izuna, Izuna gave his eyes willingly. So we can conclude that Madara was telling the truth, at least in this case.

For access to the Japanese data book I reference: https://www.reddit.com/r/Naruto/comments/2l976c/spoilers_the_complete_4th_databook/

For the translation:
https://aminoapps.com/c/anime/page/blog/naruto-data-book-izuna-uchiha/WltX_uRVgmj1rLGwaPw7envXJ0MEjo