This is just the way that nature behaves. There's nothing in the structure of electromagnetism that prevents magnetic monopoles from existing.
(If you did find a magnetic field that was structured like that of an electric dipole, then you would have a nonzero incoming and outgoing magnetic flux over closed surfaces around each of the two poles, which would imply that those two surfaces contained magnetic monopoles of opposite signs.)
Moreover, there are some important considerations which mean that finding even one magnetic dipole, in conjunction with quantum mechanics, would have strong consequences for the entire theory, and it would go a long way towards explaining some deep questions like "why does electric charge only ever come in multiples of one single elementary charge $e$?". Because of that, there has been a considerable amount of research time and money devoted to looking for magnetic monopoles.
But the fact of the matter is, every time we've looked, we've come up empty-handed. If we ever do find them, rest assured that you will hear about it, very quickly indeed. Until then, non-speculative descriptions of electromagnetism will continue to contain the statement "there are no magnetic monopoles", formalized as the Gauss law for magnetism, which forbid the types of fields you're asking about.