The sound is very rich from a tiny speaker and a very small box. I haven't a clue how they achieve so much sound from so little
I would expect the chip contains a WAV file and simply plays that back.
Also, how do they keep the battery drain so low considering the receiver must be on all the time.
My guess would be an extremely low-power circuit in the receiver which only activates the rest when it sees a square wave or whatever coming from the doorbell.
The doorbell can activate and latch itself on when pushed, send a signal for a second or so once pushed, before dropping to almost zero current again. That gives the doorbell long life because, even though active for a fair time, only activates when pushed.
The long duration of signalling which isn't obvious with a short jab on the doorbell will allow the receiving unit to sleep most of the time, waking up to check for the signal, sleeping again if not seen. Sleeping 99% of the time would give almost a 100 fold increase in battery longevity.
I am not sure how much the actual receiver itself would draw but I can't imagine it would be much, might actually have opportunities for energy harvesting.
Low-cost can be had from mass manufacturing where expensive R&D can be spread to almost nothing per unit across the quantity sold. Mass success often comes from accepting those up-front costs in order to become market leader. Many things like digital photo frames, clocks, and I would guess doorbells, are 'all the same' but with different branding.