MP3 audio standard uses "subband coding technique".
The processing at encoder end of an MP3 involves passing the input signal through a subband filterbank containing $32$ filters. Each of the filtered outputs are downsampled by a factor of $32$ (we will keep down-sampling part for later). The passband bandwidth of each of these filters is same and it is non-overlapping.
Say $Fs$ represents operating sampling frequency then each of the 32 filter banks have a bandwidth of $\frac{(\frac{Fs}{2})}{32} = \frac{Fs}{64}$.
The impulse response of each of the filters in the filterbank is obtained by using "cosine modulation technique" as described below.
The baseband filter with bandwidth $\frac{Fs}{64}$ has passband in the range $[-\frac{Fs}{128}, \frac{Fs}{128}]$. The frequency of cosine signal used by modulation to get impulse response of $k^{th}$ filter will be $(k.\frac{Fs}{64} + \frac{Fs}{128})$
Say $h[n]$ represents the impulse response of the base band signal. Then let $h_k[n]$ represent the impulse response of $k^{th}$ filter in the filterbank. The bandwidth of the base-band signal. Then, $$h_k[n] = h[n].cos(\frac{(2 . \pi . (k.\frac{Fs}{64} + \frac{Fs}{128}) . (n + D))}{Fs})$$
On simplification
$$h_k[n] = h[n].cos(\frac{(2 . \pi . (k + 1/2) . (n + D)}{64}))$$
The additional $D$ term is needed for perfect reconstruction scenario. In this case the value of $D$ is $-16$
Let $y[k][n]$ represent the output from $k-th$ filter of the filter bank for input signal $x[n]$. Then, $$y[k][n] = \sum_{i=0}^{L-1}h_k[i] . x[n-i]$$
The output from each filter will be downsampled and used for further processing. The downsampling factor being $32$ in this case. So the above equation becomes
$$y[k][32n] = \sum_{i=0}^{L-1}h_k[i] . x[32n-i]$$
$$y[k][32n] = \sum_{i=0}^{L-1}h[i].cos(\frac{(2 . \pi . (k + 1/2) . (n + D)}{64})) . x[32n-i]$$
Replacing $D$ with $-16$
$$y[k][32n] = \sum_{i=0}^{L-1}h[i].cos(\frac{(2 . \pi . (k + 1/2) . (n - 16)}{64})) . x[32n-i]$$
From the above equation it can also be observed that the cosine term in the summation is in modified form of the cosine term in DCT equation. With some rearrangement (skipping this part) of the above equation, it can be shown that the filterbank can be realized using MDCT. Thus MP3 uses MDCT to conveniently carry out filterbank processing.
Some of the reading materials online on MP3 directly keep the MDCT block in the processing steps skipping this derivation. This creates confusion for the readers about MDCT in MP3.
In summary MP3 uses MDCT only to realize the filter bank in an easier way.
AAC is another MPEG standard where MDCT is actually applied on the input signal for processing.
Hope this answer helps!