Feedback does not necessarily have to be in phase with the input signal.
The important thing to know is that a particular tube stage (or group of stages) can supply a given amount of current or voltage gain/amplification. If a signal is taken from a later stage of the circuit and applied to an earlier stage of the circuit, it is feedback. If that applied feedback reduces gain, it is negative feedback. Positive feedback increases the gain of a stage, like a microphone being pointed at a speaker producing a squeal.
So back to the cathode follower. The input signal is applied to the tube grid. This grid is normally kept somewhat more negative than the cathode, in order to bias the tube. When the tube is self-biased (or cathode-biased), then the current flowing through the tube at idle passes through a resistor connected between the cathode and ground. This current creates a voltage drop across the resistor. This voltage drop places the cathode at a voltage more positive than the grid (or you could say the grid is more negative than the cathode). If the grid were at the same voltage as the cathode (that is, if there is no difference in voltage between them), the tube becomes a diode and tube current rises to a maximum, determined by plate voltage. If the grid is made very negative compared to the cathode, then current will stop flowing through the tube, and it will be turned off.
In a typical common-cathode gain stage, there ia a large value load resistor between the tube plate and the power supply. In the cathode follower, the plate is connected directly to the power supply, and the load is moved between the cathode and ground.
Let shift gears for a moment and think about what happens if a signal is applied to a tube stage with a resistor between cathode and ground. This could be a common cathode stage or a cathode follower; the same effect will result. We assume that there is no cathode bypass cap.
Current is flowing through the cathode resistor, creating the voltage that is biasing the tube (and also determining how much current is flowing). An a.c. signal is applied to the grid. As that signal becomes more positive, the grid is being made less negative, and more current flows through the tube. This increased current creates a larger voltage drop across the cathode resistor (dictated by ohm's law, voltage=current*resistance), which makes the grid appear more negative, turning the tube off. On the next half cycle, the input signal takes a negative turn which lowers tube current. That lower current means less current is flowing through the cathode resistor, creating less voltage drop across the cathode resistor (making the grid seem more positive), which turns the tube on harder and causes more current to flow through the tube.
If you take a moment to let that all sink in, you'll see that the current flow through the cathode resistor is counteracting the change in current caused by the input signal. This changing current is what creates the voltage gain when it passes through the plate resistor, and is tapped off and passed to a subsequent stage. The phenomenon taking place is in reaction to the input signal, so it is feedback, and it is reducing gain, so it is negative feedback.
This negative feedback happens in both cathode followers and in common-cathode gain stages with unbypassed cathode resistors. The amount of feedback is smaller with the common cathode stages, because the cathode resistor is a smaller value (remember ohm's law above?).
With the relatively large value of load resistance at the cathode of the cathode follower (and no plate resistance), there is 100% feedback applied to the stage. It is called "local feedback" because the stage and its feedback are all contained in the same circuit, and not tapped from a much later stage and brought back to the earlier stage (like you could readily pick out on a schematic).
The thing that created the confusion was the "in-phase" bit. In this case, the feedback is caused by the input signal and its effect on tube current. The signal cause a change in tube current, and that changing current creates the feedback. You could probably make an argument that that means they are "in-phase". But the changing current creates a changing voltage drop across the cathode resistor, which acts in opposition to the changing voltage at the grid. The input signal and the changing voltage at the cathode are indeed in-phase, but the net effect is that stage gain is reduced, so we have negative feedback.
Gain of the cathode follower is actually less than unity, and the output might typically be 0.9-0.98 times the size of the input. The reason to use it is that the output impedance is very much lower than typical for a common cathode stage, and also that distortion caused by the tube with the unbypassed cathode resistor is reduced. In fact, you may see common cathode stages that are unbypassed, partially bypassed (both of these are very common in hi-fi), or bypassed with a capacitor chosen to restore ful stage gain only at selected frequencies as a way to tailor frequency response (overwhelmingly the norm in guitar amps).
There are formulas to go along with a lot of this (voltage gain, amount of feedback, input impednace, output impedance), but have been omitted to make things clearer (you also need access to tube data sheets for the formulas to be useful, and they are mainly close approximations anyway).
