what is the purpose of the analog part in the DAC

Yes all DAC chips output analog signals but they're often not in the right format for sending to a preamp.

Actually, this is incorrect.  DAC chips only output DC (direct current) pulses that are at different levels.  The "I/V stage" is an analog stage and it is required at a minimum.  The job of the I/V stage is to take those DC pulses as an input and help to try to form that into an analog waveform.  The byproduct of slow slew rate and voltage increase with this signal amplification stage is used to help that.

There can be other analog stages, like a low-pass filter stage that works to remove those ultrasonics.  An output buffer stage may also be added to help drive the input impedance of the target device (i.e. preamp or amp).

...to make sure the receiving unit can work properly with the dac chip’s output, which is often too weak to drive the next piece of hifi equipment

https://en.wikipedia.org/wiki/Impedance_matching

https://www.youtube.com/watch?v=fsvmJ_x1t3o

It filters, isolates and amplifies the output from the DAC to a usable level. Typically, the better the analog stages are, the better the overall sound will be.

The "I/V stage" is an analog stage and it is required at a minimum. The job of the I/V stage is to take those DC pulses as an input and help to try to form that into an analog waveform.

This isn't accurate. Not all DAC chips need an I/V stage, only those whose internal circuits work in current mode. AKM chips (excluding the AK4499) work in voltage mode and use capacitors as DAC elements so they output a voltage directly but normally balanced.

The DC pulses out of a DAC are already analog in nature, that's the basic function of a D/A converter.

The real sound is made outside the chip by power supplies, filters, clocks, and analog stages.  I think all too much attention is paid to the D to A conversion.

As a vinyl veteran I like to think of the digital to analog converter (be it DS chip, multibit chip or R2R circuit) as the turntable, arm and cartridge.  The analog stage is analogous to the phono pre.

Analog output stages generally include:
1) I-V conversion (current to voltage)
2) Low-pass filters
3) Gain stage
4) Output buffers

Even in some very expensive DACs too little effort is provided to the power supplies and analog stages IMO.  Cheap  switching supplies and cheap chip op amps are used even in some high cost DACs.  Vinyl addicts would think twice about allowing that to happen in a costly phono pre.  I think it is what keeps some lp addicts from enjoying music from a DAC.