It isn't that Bitcoin solves a different problem it is that is solves it better.

How many merchants accept Bitcoin, how many accept Doge?  How many well run, highly capitalized solution providers like BitPay are building infrastructure around Bitcoin?  How many around doge?  Building large scalable businesses requires capital.  How many private equity offering for companies that build Bitcoin related services have happened vs doge?  I think you get the point.

There is something called the network effect and no passing a few pennies between users isn't a network effect.  Bitcoin is still in the infancy but the number of merchants willing to accept it is growing.  As long as it is growing the network effect will grow with it.  When an Overstock, Wordpress, or even Kongregate starts accepting Doge you might have a point.  Does this mean Bitcoin is the one currency to rule them all? No.  I think it is possible superior alternatives will either co-exist or replace Bitcoin but they will need to be truly superior alternatives.

Your logical fallacy would be like me inventing a totally incompatible cellphone called a dogephone and this dogephone works just like a cellphone except the only people you can call is other people who have a dogephone.  Most of the people on the planet, have no interest in getting a dogephone because their cellphone works just fine.  When you point that out to me I say "what problem does a cellphone solve that Dogephone does not?"  It is a logical fallacy because while the answer may be none, it is a completely meaningless answer.