Let's say a Wizard is outside the range of a 3rd Level Shadow Monk's Silence, casted via Ki points.

Can he dispel it?

My gut inclination is that the Wizard can indeed dispel the Shadow Monk's Silence despite it not being technically a spell, due to the Sage Advice Compendium containing this ruling:

• Is it a spell? Or does it let you create the effects of a spell
  that’s mentioned in its description?
  

[...]

If any one of these questions can be answer "yes", then the effect, ability or item is magical for the purposes of being affected by magic cancelling effects.

Now, consider the Shadow Monk's Shadow Arts Feature:

Shadow Arts

You can use your ki to duplicate the effects of certain spells. As an action, you can spend 2 ki points to cast darkness, darkvision, pass without trace, or silence, without providing material components. Additionally, you gain the minor illusion cantrip if you don't already know it.

Seems clear enough. However, the only reason I'm unsure is due to to the "best answer" selected to this question, "Does ki count as magic for the purpose of an antimagic field, or is it only fluff?"

That answer seems to conclude that Ki Magic is NOT in fact magical. (Note that the second answer, which is more highly rated, concluding that it IS in fact magical.)

I'm likely to face this specific, niche situation in an upcoming session I'm running and I want a clear, community consensus on how to resolve this specific problem. Other examples and evidence are extremely welcome as well.

Yes, you can dispel it

Shadow Arts

Starting when you choose this tradition at 3rd level, you can use your
ki to duplicate the effects of certain spells. As an action, you can
spend 2 ki points to cast darkness, darkvision, pass without trace, or
silence, without providing material components.

The key term here is "cast" because it means you are dealing with an actual spell. The ability simply allows you to cast a spell. It even talks about components, something that is only relevant if you are dealing with actual spells (not "spell-like" abilities). And dispel magic works against spells.

Yes, that first line is a bit deceiving because you'd expect the thing coming after to be duplicating the effects of the spell only and not be one, but the language after it is pretty unambiguous that you cast an actual spell.

If you want to dive deeper into what things count as spells or not you can take a look at my answer on What counts as a spell?

The whole issue with ki being magical or not is also a red herring here. Dispel magic (despite its name) only works on actual spells and doesn't care if something is technically magical or not.

Therefore, dipel magic will indeed work on any of the shadow monk's Shadow Arts spells.