I've discussed this to some degree here, but the main point to take away is that we should not use the manga and anime tags to denote a question about a specific adaptation.
With that said, there are two situations that need to be considered. The first is when the manga and anime only differ slightly, such as the Bleach and Naruto animes. Although they add in additional spoilers, and sometimes remove things, they follow the same overarching plot as the manga. The second situation is when the manga and anime are significantly different, such as the the Full Metal Alchemist manga and the 2003 anime. Although they are both based in the same world, and share many of the same themes and characters, the plot is completely different. They are essentially two different, if not somewhat related, works.
For the second situation, where the anime and manga differ significantly then they should both have separate tags. The only connection these works share is being part of the same series. Having separate tags for these works will make it easier for users to find answers to their questions. After all, if I'm asking a question about the 2003 FMA anime, then I'm really not that interested in questions about the Manga or Brotherhood since the series are drastically different. It would be a lot easier if I can just filter my search results to only include questions about the 2003 FMA anime. We want to make things easier for the user, not more difficult.
In the situation that the manga and anime are similiar, then I think stating which adaptation you're talking about in the answer is enough. For the most part the answers will be the same, with the most notable exception being fillers. Having two separate questions, one for the anime and another for the manga seems fairly pointless. All it will do is make it harder for users to find answers to their questions. Remember, we want to make things easier for users, not more difficult.
Something else to consider is that a question belongs to the community as whole, not just the asker. Just because the OP wants an answer only relevant to the manga doesn't mean that's the only answers we allowed. The OP would accept the answer that best answers for the work he's interested in, and users would just vote on the other questions accordingly. That way anyone with the same question, bot interested in an answer based on the other adaptation, can find their answer in the same place.