Often in spoken English, people say, "So next day I went to the bank". They just drop "the"

When 'next' is used as an adjective it modifies the noun after it. Since 'next' normally refers to something specific it is nearly always preceded by 'the'. ' There are drinks in the next room. (Which room. the next one) 'They met again the next day.' (Which day? the next one). However we don't use 'the' before 'next' before names of days (Monday etc), months (January etc), the words 'week' and 'year' unless we are referring to a specific day, week, month etc. 'The next Sunday it was raining.' So if you see 'next day' without 'the' it is a grammatical error but people sometimes omit 'the' in informal English.

The correct phrase is "the next day". I don't know the phrase "next day".