Toilet seats are components that are frequently replaced with wear and time. You may not need to replace them as often as you replace the rubber components inside the tank, but you should reasonably expect to replace them every few years, to keep a new appearance.
The discoloration could be the sun, but it could also be caused by the cleaning products used on it.
Toilet's typically don't come with toilet seats, unless they are non-standard. So the builder/plumber buys the toilet seat separately when installing a new unit. There are two standard type of seats, "round" and "elongated". What you have appears to be elongated. Measure front to back, and take a picture of your measurement and go to your nearest home improvement center (Home depot, Lowes, etc) and ask for help selecting a replacement. Replacements are inexpensive, with the "better" ones having a quiet close feature. Be sure to match your toilet's color, if your toilet isn't white (not all are white). Installation is fairly simple, and requires basic hand tools.
Your landlord could replace this for you, but that is really up to them, as their responsibility is that it works, not that it is pretty. To make the home pretty is your responsibility, if you don't think the landlord is doing enough. The landlord might be willing to install it for you if you buy it... That would depend on them, and their relationship with you. This is an inexpensive repair, but be aware that it becomes the landlord's property legally, so don't try to take it with you when you leave (unless you put the old one back, but don't save that unsanitary old thing, toss it in the garbage).
I replace toilet seats as a move-in practice, and I replace them again as soon as they show any signs of wear or general aging. I buy nice whisper-close units and pay around $40, but very cheap ones are also available. Just keep in mind that the cheaper you go, the more likely you will replace it again.