Let's go back to the original fundamental definitions of the terms.
"Analog" mean that there is an analogy between two things. For example, in a mercury thermometer, there is a relationship between the height of the mercury in the thermometer tube and the outside temperature. Or in a car tachometer, there is a relationship between the angle of the pointer on the dial and the speed of the engine. Or in an analog clock there is a relationship between the angle of the two hands and the time. A very common type of device is where there is an electrical voltage and there is a relationship between the voltage and something else. e.g. the voltage is related to the temperature or the engine rpm. Bottom line: analog simply means that one thing is represented by analogy to another thing.
"Digital" means that a thing is not represented by another thing, but directly by a number. So in a digital thermometer, we don't have a level of mercury in a tube, we have a number. In a digital tachometer, we don't have a pointer on a dial, we have a number. In a digital clock, we don't have hands we have a number. So in digital electronics, we don't have a voltage that is proportional to a quantity, we have a number. That number is usually commonly encoded in binary, although it could be something else, like balanced ternary.
Now, "continuous" means that given any two values, there are an infinite number of intermediate value. E.g. like the real numbers. In between 1.0 and 1.1 there are an infinite number of intermediate values.
"Discrete" means that there are some values which do not have any values in between. e.g. like the integers. There is no integer between 1 and 2.
Now, continuous and discrete may refer to either time or the quantity being measured. e.g. you can have continuous time systems, discrete time systems, but you can also have a continuous valued waveform or a discrete quantized waveform.
So, strictly speaking, analog does not equal continuous and digital does not equal discrete. However, in common usage in the context of digital signal processing, analog systems are usually continuous in time and value, and digital systems are usually discrete in time and value. So the terminology does get abused somewhat.