I've been trying to learn piano by myself for a while, and I've seen many people say that I won't be able to learn the piano properly if I don't have weighted keys. And I can't buy another keyboard. Can I learn piano using a non-weighted keyboard?
Strictly speaking, no, you can't. However, it will get you a fair way into what the black and white bits do: you'll be able to play tunes, chords, harmony, and learn to play in different keys, just like piano, all of which will set you up for when a piano comes along. You may even be able to use a sustain pedal, rather like a proper piano.
What you won't be able to do is get the feel of the proper weighted keys, which is all part of the expression side of piano playing. But, using an organ type keyboard will get you maybe 75% of the way there. You may or may not pick up bad habits on the way, and certainly your playing technique won't be good for real piano playing, depending on how good (or bad) your keys actually are. There's more to be gained than lost, so go for it!
Practising using piano sounds on an unweighted keyboard is, in a manner, the worst of two worlds: it can only prepare you very sketchingly (and that's assuming that the non-weighted keys are at least velocity-sensitive) for the graded attack of a piano, and piano sounds are useless for practising the legato control necessary for playing organ, harmonium and other continous-sounding instruments convincingly.
You'll play piano like an organ player and organ like a piano player. Which is better than playing piano like a violin player, granted.
For example, your trills are likely to suck badly when you sit down at either an organ or a piano for real.
So if you are not going to be able to afford a piano anyway, what is your interest in the instrument based on?
Yes, you can "learn piano" using non-weighted keys; HOWEVER, your fingers will not develop the strength needed to perform smoothly and with good dynamic control.
It depends partly on what kind of music you are going to be playing the most of, down the road. If classical, you won't be able to progress very far beyond just the rudimentary pieces. Jazz, ditto. But if your goal is to be a member of a stage band, you may be OK, depending on the complexity of the music you try to play.
In my more than fifty years of piano teaching I have known several students (out of hundreds) who were able to achieve excellent technique while playing primarily on non-weighted keys, but it's rare. I always recommended that my students practice on weighted keys if at all possible, but, as in your case, it wasn't always possible.
Good luck, and don't give up. With motivation to learn, you will.