As others have stated, "zoom" is a ratio, not a measure of focal length or angle of view. What the zoom ratio does tell you is how much narrower the field of view (FoV) will be at the longest focal length setting when compared to the shortest focal length setting. Another thing it normally tells you is the degree of unwanted things like distortion, chromatic aberration, and poor resolution you can expect from a particular zoom lens. The larger the zoom ratio is, the more optical compromises have to be made to allow the lens to operate over such a wide range or the more corrective elements must be included in the lens' design (and weight, size, and price) to counteract them. This is especially applicable in the case of lenses designed for larger sized sensors such as those found in DSLRs and other interchangeable lens cameras.
Digital zoom is just another way of taking a photo and trimming parts of it to leave what is in the middle of the image. If you have a 12MP sensor and use 2X digital zoom, what you have done is use only the 3MP at the center of the sensor and thrown the rest away. Remember Megapixels are computed using area, not linear measurements. If you use only one half of each side of a sensor, the area is one fourth as large.
You are correct that a focal length twice as long (2X zoom) as the original one will make objects appear twice as tall and wide in a photo taken from the same location. The perspective of the picture taken from twice as far with 2X zoom will be different than the one taken from half the distance and the same focal length, though. The longer the focal length you use, the more the distance between closer and further objects will be compressed, making it appear there is less space between them than there really is. On the other hand, a wide angle lens will expand the apparent distance between objects different distances from the camera.