Doppler effect and the twin paradox
There are two different kinds of doppler effect. The one you are describing is the normal one which applies to waves that travel at a fixed speed in a medium independent of the speed of the sources of the waves and the destinations of the waves. The other one is called relativistic doppler which is a combination of the first one, plus the reciprocal effect of time dilation between two observers, which results in the doppler frequency being independent of any medium.
When we are talking about the twin paradox, we are assuming that each one has an identical clock that ticks at the same rate when they are at rest with respect to each other and that they can see the ticking of each other's clock; think of a clock that flashes a light once per second. When we put them in relative motion, it takes time for the once-per-second light flashes to travel between them (they are both sending out flashes and observing the other one's flashes).
Now if there were an absolute medium to carry the flashes between them and if there were no time dilation, then we could think of one of them at rest in this medium and the other one traveling away through this medium.
Let's say that the traveling twin is going at 90% of the speed of light. What will have happened after ten seconds? Well, the stationary twin will have emitted eleven flashes (counting the first and the last ones) but the traveling twin will be 9 light-seconds away and will just now be seeing the flash that was emitted 9 seconds earlier, which is the second flash (the first one he saw just as he was leaving). The other nine flashes are still trying to catch up to him.
In the mean time, the traveling twin has also emitted eleven flashes but a lot of them have already been seen by the stationary twin because they were emitted when the traveling twin was much closer to him.
So the doppler frequencies for the two twins, assuming a stationary medium to carry the light and no time dilation, are different. By doppler frequency, we mean the measured rate at which the flashes are detected by each twin of the other twin's clock flashes compared to their own clock tick rates (which are the same in this pretend example).
But, we can't ignore time dilation and what happens in reality, is that the traveling twin's clock slows down, it ticks more slowly than the stationary twin's clock. This has two effects: first it means that the traveling twin will percieve the stationary twin's clock flashes to be coming in faster because he is comparing them to his slow clock and, second, it means that he will be emitting his flashes at a slower rate, and so the stationary twin will see the traveling twin's clock flashes coming in at a slower rate. It turns out, wouldn't you know, that these two effects result in each twin seeing the other's clock as ticking at exactly the same slow rate compared to their own.
And it turns out, wouldn't you know, that no matter how fast the traveling twin is going compared to the stationary twin, they will both observe each other's clocks as going slow compared to their own. As a matter of fact, we could have both twins traveling in opposite directions at the same or different speeds and still, they would each observe the other's clock going at the same slow rate compared to their own.
What this means is that the relativistic doppler is independent of the relative speeds of the observers with respect to any medium. Relativistic doppler depends on only one speed, the relative speed between the two observers and is identical for the two observers, whereas the normal kind of doppler is dependent on two speeds, the speed of each observer with respect to the stationary medium and produces two different doppler frequencies, one for each observer.