Let's take this example. 1/ There you are, the VERY person I was looking for. ->> you are the exact person I was looking for. 2/ The VERY next day. -> It's the day next to the day you say this, it's not 2 or 3 days later. So "very day" can be understood that it's exact the day you mention. In conclusion, VERY is used for emphasis.

It's a somewhat old-fashioned usage, but you can use it in something like 'that very day', where 'very' means 'exact(ly)', which is the same idea as Gertie's example. "He chose that very day to arrive". "on this very day, the planets are almost aligned".

"Very day" isn't a word/correct. "Very same day" is the same as "same day". "Very" just adds emphasis to "same".