The intention is obligatory for each day of Ramadan because each day is an independent act of worship separate from the others.

The intention must be made at night before the break of dawn, as the Prophetﷺ said: "Whoever does not intend fasting at night, there is no fast for him." [An-Nasa’i] 

And he also said: "Whoever does not firmly resolve to fast before dawn, there is no fast for him." [At-Tirmidhi, Abu Dawood, and An-Nasa’i]

Whoever wakes up and eats Suhoor while mindful of fasting has made the intention. Likewise, one who firmly intends at any moment during the night to fast the next day has also fulfilled the intention.