The phrase "gives me a ..." indicates that we are talking about a reaction to an emotion like fear or horror. In that context they are synonymous. We can say "a chill"/"a shiver" but it is common to pluralize them: "it gives me the chills"/"the shivers." A "chill" is a sensation or feeling. A "shiver" is the way your body responds to that feeling. (Another way is with "gooseflesh.") A "chill" emphasizes temperature. It means feeling cold. The most common reason for feeling cold is that it really is cold. "Don't go out without a coat, it's cold today, you'll get a chill." Or "You'll get chilled." Or "It's chilly today." A "shiver" is an involuntary muscle trembling. A common reason for shivering is a response to cold. Shivering creates heat in the muscle and actually warms you. "Gooseflesh" is another reaction to cold: the skin gets bumpy. However, we can feel an illusion of cold, and we can shiver or feel an illusion of shivering. "When I look at that rollercoaster, it sends shivers up and down my spine." A song from a musical says "When shivering in my shoes... I whistle a happy tune so no one will suspect I'm afraid."

They have the same meaning, but chills is more common. Usually it’s said like this: “It gives me the chills.” 🌈 🫶🏼

(A) It gives me a chill. ^this is grammatically correct but you'll probably not hear a native speaker say this because it feels unnatural. A more common/natural way to say this would be to pluralize the direct object: "It gives me the chills" which implies that something "freaks you out" or makes you very uncomfortable. It could also be used to convey something very emotionally moving or powerful, like recently I watched a video of a deaf child use new technology to hear his parents speak for the first time, and that gave me chills. I had goosebumps on my arms and tears in my eyes. A singular "chill" is rarely, if ever used, and would typically be a reference to temperature, like "it's chilly in here". (B) It gives me a shiver. Same thing here, you'd need to pluralize this for it to really sound natural: "It gives me the shivers", or "I felt a shiver down my spine". I think shiver is less flexible than "chill" because I typically think of it as a negative sensation.

In my opinion, no. Technically a shiver could be thought of as a more sudden, sharp "chill." When I think of shivering, I think of someone shaking from the cold. When I think of a chill, I think of fear (The ghost story gave me chills) or sickness (I had the chills when I got the flu.)