Is the DAC the digital equivalent of a cartridge.

There is no analogy to cartridges. All cartridges are different and impart a different taste/color to the modulations in a record.

Digital music is perfect. All DACs are the same. They are incapable of imparting a taste, or color to the digital files. Because digital music is perfect. 

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.