In spin ice systems magnetic monopole-like excitations are sources or sinks of $H$, not the $B$ field, why is that? Is it because the strings carries magnetic moment $M$ and not solenoidal $B$ filed like the Dirac string.
How is it different from Dirac construction of magnetic monopole?
In the Dirac case we still have incoming flux from the solenoid and out going from the monopole that will result in 0 for any sphere around the end. How he goes around this (why is it justified to not count the singularity) and why is not possible for the spin ice case?
A magnetic monopole for B (not H) is the situation discussed by Dirac, 't Hooft, and many other particle physicists, and that's the situation which would be interesting for various fundamental physics reasons.
For 75+ years, physicists were using the term "magnetic monopole" to refer to a very specific thing: Magnetic monopoles for B. People had analyzed them theoretically and searched for them experimentally. Whole books were written about them.
Then in the 2000s, certain scientists discovered phenomena in spin ice and other systems that had nothing whatsoever to do with that, but nevertheless they made the decision to call these things "magnetic monopoles". Pause to think about this. For 75+ years, "magnetic monopoles" was a term with a specific technical meaning in physics that everyone agreed on ... but then these scientists used the same term to describe a completely unrelated phenomenon. Why would they do this?? Um, well, presumably, they intended to mislead gullable science journalists and the general public into thinking that they made the long-awaited discovery of "true" magnetic monopoles (for B). The spin-ice and other experiments were legitimately interesting and exciting works of condensed-matter physics, but it sounds much more exciting when it is described in this extremely misleading way! [I don't know their actual thought process, I'm speculating.]
Well, if they were trying to mislead people, they were successful! In many many popular-science articles in the weeks and months following this work, journalists confused the spin-ice (so-called) monopoles with the (true) magnetic monopoles. In the wikipedia article, which I was monitoring at the time, well-intentioned readers changed the article again and again to say that the Dirac monopole has now been discovered.
To be clear: Electrons are not (true) magnetic monopoles, i.e. they are not sources or sinks for B. The same is true for protons and neutrons, and all known particles. Therefore if you put together electrons and protons and neutrons into any possible configuration, no matter how many there are or what the configuration looks like, it is mathematically impossible to create a source or sink of B. Therefore a "true" magnetic monopole will never ever be created using condensed-matter physics wizardry.
It seems to me that the right starting point is to say that the Dirac monopole and the spin-ice "monopole" are two completely unrelated phenomena in different branches of physics. They were only mixed up due to false advertising. If you have a question about Dirac's discussion of magnetic monopoles, you should pose it as a separate stackexchange question, and likewise if you have a question about spin-ice monopoles, you should pose it as a separate question. I will try to summarize quickly:
The "Dirac string" isn't a real thing, it's merely a mathematical construction. We expect and demand that the string is unobservable, because in reality there is no string. On the other hand, a spin-ice "monopole" has an actual "string" - a chain of flipped spins - that is an observable physical object.
To prove charge quantization from a Dirac monopole, you start with the assumption that the Dirac string is unobservable (in one mathematical approach). If the string is a real physical string that can be observed, the argument doesn't work -- so charge does not have to be quantized.