In my experience, no! I actually wondered this myself. I started one in Georgia (did not turn out well :') ). Tasted very different from the one I bought here, which I also had in Georgia. I went west to California. My roomie went north to Mass. Hers got moldy and trashed eventually, but still tasted the same. Mine still tastes the same way it did in GA, a few thousand miles away. It's been a few years now. Flavor hasn't changed much.
From a science standpoint, the symbiotic culture of yeast and bacteria of a well established starter should be robust enough that environmental microbes are outcompeted and can't get established. This is also why a healthy starter will never mold even though it's a heap of wet nutrients sitting on your counter (imagine if you left out a soggy loaf of bread, which is pretty much a starter without the culture, and how fast it would mold).
All locations on earth have their own unique localized combinations of yeasts and bacteria floating about in the air. Now add in that all flours also contain their own combinations of yeasts and bacteria from the grains and the mills. Now add in humidity and ambient temperatures the encourage or discourage competing strains of yeast and you can begin to see the randomness and varieties that have evolved. Essentially, if you start your own Sourdough starter in your home there is still a very good chance that it may still differ from your next door neighbor's due to the myriad of variables. So settle on a Starter that you like and nurture it as your baby. Hope that helps.
Doen't the starter conform to the local environment after a feww weeks of using and feeding?
All locations on earth have their own unique localized combinations of yeasts and bacteria floating about in the air. Now add in that all flours also contain their own combinations of yeasts and bacteria from the grains and the mills. Now add in humidity and ambient temperatures the encourage or discourage competing strains of yeast and you can begin to see the randomness and varieties that have evolved. Essentially, if you start your own Sourdough starter in your home there is still a very good chance that it may still differ from your next door neighbor's due to the myriad of variables. So settle on a Starter that you like and nurture it as your baby. Hope that helps.