How much of a difference would it make to the final product and which one would be the better choice for any enriched sourdough breads/pastries/cakes etc? And by "regular starter" I mean the normal flour and water mixture. Thanks!
If it can be done, it will make lots of difference. Different sourdough recipes are geared towards breeding different bacterial species, which give their own taste to the final product.
There is no answer to "what will be a better choice". If you have a known good recipe for a starter using milk, then it is up to the eater to decide which result tastes better to them. It's literally a matter of taste.
Note that, if you just thought to pour some milk and flour together, this is a terrible idea. Fermented foods are only safe when following the recipe to the letter. A small change in the initial conditions can breed a completely different strain of bacteria - which can be simply yucky, but can also be seriously dangerous for your health. So if you don't have a trusted source which describes the exact way to produce a safe sourdough starter by using milk, don't experiment on your own.