LPF in the stage of IQ demodulator is it a analgor filter or digital filter?
The A/D can be placed as a single real A/D before the multipliers, OR as shown in the diagram as two A/Ds one after each multiplier to sample the I and Q channels. In either case, an analog filter is required before any A/D conversion as an anti-alias filter. This can be a bandpass filter or a low-pass filter, depending on which image in the analog domain's frequency spectrum is desired as the signal of interest. Even if there is no actual additional signal in the image bands, with no filter (and gain in the receiver) the noise figure will be significantly degraded due to the accumulated noise floor from each of the images. An analog filter prior to any sampling is important.
So in this case given the placement of the A/D converter, the filters as shown would be the analog anti-alias filter that I describe.
If the reason for image rejection is confusing, please refer to this posts that details how aliasing occurs in the digital sampling process: Higher order harmonics during sampling
So in summary, for the implementation of an IQ demodulator, there could be two A/D converters if I and Q are to be sampled separately: one for the I channel and one for the Q channel resulting in a complex representation of the baseband signal. Alternatively there can be a single A/D converter prior to the multipliers and then the multiplier and filter structures as shown can be implemented all digitally (referred to as a Digital Down-Converter or DDC). In this case, a single analog anti-alias filter (band-pass or low pass as desired for the implementation) is still needed in front of the A/D converter, in addition to the digital low pass filters after the multipliers to select the low frequency component after the multipliers.
The block diagram as shown in the question does not make that clear that $v(t)$, $y(t)$ and $\hat d[k]$ are complex signals (although clearer if you read through the linked reference).