Is the DAC the digital equivalent of a cartridge.

The transport is the equivalent of the table, arm & cartridge combined. They both spin a disk hopefully at an accurate speed, align a device to “ read” the disk, read it & send that  information on its way. The phono preamp & the DAC are then sort of equivalent in that they take the signal they receive and make it useable to a pre amp or possibly a power amp itself. 

Though the cartridge is the closest to the media, other components in the readback system is as, if not, more important.

- The turntable- to present the media to be read at the correct and precise speed. Insulated from external (environment) and internal (motor) vibrations.

- the tonearm-to support the cartridge smoothly across the vinyl without hindrance and insulated from extraneous resonances.

The above two needs to do a good job for the cartridge to give of its best.

BTW, the dac box is the phonostage.
Cause that’s where line level signal is outputted.

The phono stage just takes takes the information from the cartridge and amplifies it to line level; however, the DAC actually is in charge of decoding/converting/ reading the digital information that is fed into it.  The DAC does no amplifying to the signal as far as I know.  

I know all carts have a different sound, and the range of is crazy.  Carts have a huge effect on the sound, I just want to know if the DAC has as big an effect as a cart  

Sorry, the phono stage does not decode anything, all it does is boost/amplify the small electric signal given to it by the cartridge. No way is a DAC anything like a phono stage, there’s no boosting or amplifying in a standalone DAC.

If you feed a DAC a good digital signal: streaming, cd, files, etc.. it should behave exactly like cartridge does as it is fed the grooves of a record, except digital signals don''t need to be boosted because they already line level.  

Sorry, the phono stage does not decode anything

wrong. ever heard of the RIAA equalization curve? what do you suppose it does?

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

some phono stages have multiple output curves besides RIAA for optimizing various early pressings. but your records would sound like crap without any output ’decoding’ circuit. lp groove cutting realities and playback technology realities require this equalization to work.

there’s no boosting or amplifying in a standalone DAC

wrong again. the only way a dac does not have an analog output stage to amplify the dac output is if the dac chips develop enough voltage on their own (very rare) or the dac is feeding an amp inside an active speaker system (very rare). dac chips rarely develop sufficient voltage to raise the signal to line level to feed the preamp. that is what the analog output stage circuit inside the dac does.

maybe the dac in your phone might develop enough gain to power your ear buds, as everything is miniaturized. but likely it has an output stage of some sort too. but it’s a special case.