Can I ask for a "general" letter of recommendation?
No, you can't. Here are some reasons that this would not be a reasonable request [in the US].


Your professor is not going to show you the letter, so they will need to send it to every internship, job, or grad school themselves. Therefore there is no benefit to them to having a letter "on reserve".
Letters are written very differently for internships and industry positions than for grad school applications, so a general letter does not make sense.
There is no reason for a professor to waste their time writing a letter now that might not be used at all.
There is no reason to write a letter that will be outdated. It is unlikely that a letter based on a freshman class would be useful for grad school applications, for example.


Instead, what you can do is ask your professor "I don't have a position in mind right now, but would you be willing and able to write me a strong letter of recommendation in the future for internships, jobs, REUs, or eventually grad school?" If they say yes, say something like "Thank you,  in that case I would like to use your letter when I do. Is there anything I can do, now or then, to make it easier to write?" This gives them a chance to let you know what they need (what materials, how much advance notice, how much reminding); it also gives them a chance to make notes to themselves about your performance, to remind themselves when they do go to write the letter.

(You asked "...or are rec letters supposed to be tailored to specific applications?" In my field almost everyone writes just one letter for grad school applications, there aren't usually different letters for different schools; however letters for REUs or internships would be very different.)

It's wise of you to start thinking about this now, and you'll be glad to know you have someone who can write you a strong letter; they just won't write it until there is actually a position you're applying for (and not until the night before the deadline, if I'm any example).