Multiple identities in one brain/body can arguably be considered separate moral patients, whether they are naturally occurring through a brain quirk, a childhood trauma, iatrogenically induced by a hapless therapist or a malevolent cult leader, or intentionally created by the "original". 

Tulpas are not special that way.

There is a spectrum of identity consciousness and self-awareness, ranging from a vague fragment to a fully separate and conscious mind. Presumably one should give more moral weight to the identities that are more developed, but the issue is rather complicated. 

My belief is that yes, tulpas are people of their own (and therefore moral patients). My reasoning is as follows.

If I am a person and have a tulpa and they are not a person of their own, then there must either (a) exist some statement which is a requirement for personhood and which is true about me but not true about the tulpa, or (b) the tulpa and I must be the same person.

In the case of (a), tulpas have analogues to emotions, desires, beliefs, personality, sense of identity, and they behave intelligently. They seem to have everything that I care about in a person. Your mileage may vary, but I've thought about this subject a lot and have not been able to find anything that tulpas are missing which seems like it might be an actual requirement for personhood. Note that a useful thought experiment when investigating possible requirements for personhood that tulpas don't meet is to imagine a non-tulpa with an analogous disability, and see if you would still consider the non-tulpa with that disability to be a person.

Now, if we grant that the tulpa is a person, we must still show that (b) is wrong, and that they are not the same person as the their headmate. My argument here is also very simple. I simply observe that tulpas have different emotions, desires, beliefs, personality, and sense of identity than their headmate. Since these are basically all the things I actually care about in a person, it doesn't make sense to say that someone who differs in all those ways is the same. In addition, I don't think that sharing a brain is a good reason to say that they are the same person, for a similar reason to why I wouldn't consider myself to be the same person as an AI that was simulating me inside its own processors.

Obviously, as with all arguments about consciousness and morality, these arguments are not airtight, but I think they show that the personhood of tulpas should not be easily dismissed.

Edit: I've provided my personal definition of the word "tulpa" in my second reply to Slider below. I do not have a precise definition of the word "person", but I challenge readers to try to identify what difference between tulpas and non-tulpas they think would disqualify a tulpa from being a person.

Arguably there has been a lot of work done on this topic, its just smeared out into different labels, the trick is to notice when different labels are being used to point to the same things. Tulpas, characters, identities, stories, memes, narratives, they're all the same. Are they important to being able to ground yourself in your substrate and provide you with a map to navigate the world by? Yes. Do they have moral patiency? Well, now we're getting into dangerous territory because "moral patiency" is itself a narrative construct. One could argue that in a sense the character is more "real" than the thinking meat is, or that the character matters more and is more important than the thinking meat, but of course the character would think that from the inside.

It's actually even worse than that, because "realness" is also a narrative construct, and where you place the pointer for it is going to have all sorts of implications for how you interpret the world and what you consider meaningful. Is it more important to preserve someone's physical body, or their memetic legacy? Would you live forever if it meant you changed utterly and became someone else to do it, or would you rather die but have your memetic core remain embedded in the world for eternity? What's more important, the soul or the stardust? Sure the stardust is what does all the feeling and experiencing, but the soul is the part that actually gets to talk. Reality doesn't have a rock to stand on in the noosphere, everything you'd use as a pointer towards it could also point towards another component of the narrative you're embedded within. At least natural selection only acts along one axis, here, you are torn apart.

Moral patiency itself is a part of the memetic landscape which you are navigating, along with every other meme you could be using to discover, decide, and determine the truth (which in this case is itself a bunch of memes). This means that the question you're asking is less along the lines of "which type of fuel will give me the best road performance" and more like "am I trying to build a car or a submarine?" 

Sometimes it's worth considering tulpas as moral patients, especially because they can sometimes manifest out of repressed desires and unmet needs that someone has, meaning they might be a better pointer to that person's needs than what they were telling you before the tulpa showed up. However if you're going to do the utilitarian sand grain counter game? Tulpas are a huge leak, they basically let someone turn themselves into a utility monster simply by bifurcating their internal mental landscape, and it would be very unwise to not consider the moral weight of a given tulpa as equal to X/n where n is the number of members within their system. If you're a deontologist, you might be best served by splitting the difference and considering the tulpas as moral patients but the system as a whole as a moral agent, to prevent the laundering of responsibility between headmates.

Overall, if you just want a short easy answer to the question asked in the title: No.

We've all heard the idea that there exists two selves, the self that exists in your own mind, and the self that exists inside the perceptions of others.

Intentionally created 'tulpa' must be similar to the emulations of so many people I've closely interacted. The ones that exist lurking in my subconscious mind. Instantiated via my intuitions of how they'd respond to a question, or wondering what gifts they would appreciate.

How about in dream characters. Is it wrong to murder dream characters, and should we strive to lengthen dream time to give them all a longer more fulfilled life?

Even the morality of sci-fi brain emulation is murky to me. Let alone the type of emulation we all do unconsciously ourselves. I'd have to hear a very convincing argument to separate tulpas that say "hi I'm here and alive!" from dream characters that do the same thing, or other illusions like chat gtp.