Sounds to me more like a situation where they are cash-strapped. It's unlikely that people at a startup have the time and focus to plan a good internship program.
That is true. Startups do not have time to invest in interns so they must have some faith in you to bring you in.
On the other hand, a startup by students sounds a little skeptical to me. You will be working with people that do not have experience building projects at big scale or delivering them successfully to a user base. If the startup is full of students, who will mentor them ? Even mature startups that have experienced engineers and managers need mentors. Make sure they have a good direction and know what they are doing before you join. You need to derive value from them as much as they derive value from you.
Good luck.