What does the DWT do at the j-th step?

I have a theoretical question about the calculation of the Discrete Wavelet Transform, using MATLAB. According to the video tutorial:

https://it.mathworks.com/videos/understanding-wavelets-part-2-types-of-wavelet-transforms-121281.html

the DWT algorithm applies a low-pass and a high-pass filter to a signal to obtain a low frequency signal and a high frequency signal, using appropriate filters. And this makes sense to me, because the output of the application of a filter, such as a FIR filter, (i.e using the FILTER function in MATLAB) is another signal.

However, looking at the documentation of the function DWT or WAVEDEC in Maltab, it looks like the output of a j-th step of the DWT is NOT another signal, rather the cAj and cDj coefficients, from which the low frequancy and high frequency signals can be then reconstructed.

So my question is: what is the filter that is applied to the signal? Why does it provide coefficients and not another signal? What is the difference between the application of the DWT and that of a FIR filter? The mathematical formulas of the two seem analogous.

In other words, it seems that I apply a filter (with some coefficients) to find other coefficients, rather than another signal ... this is not 100% clear to me.Maybe I misunderstood some basic concepts, can you please clarify?