Ah yes, melanistic eevee. It’s actually quite interesting, to be honest. Melanistic eevee are more or less akin to albinos of the same species: not shiny but rather pokémon with melanin levels that differ from the norm. However, unlike albino pokémon, melanism in pokémon can be an adaptational advantage and—in the absence of melanism-related disorders such as ocular melanosis—often is.
That’s important to clarify because melanism is a genetic condition. In laymen’s terms, it’s caused by a mutation in a pokémon’s genetic code that causes them to have one extra copy of a specific sequence that controls melanin production. However, whether or not that mutation is expressed is a recessive trait. As such, pokémon may either carry it and pass it to their young or, well, be melanistic themselves.
As for the reason why this mutation occurred in the first place, as mentioned above, it’s often an adaptational advantage. For example, while non-melanistic umbreon use their rings’ glow as an intimidation tactic, umbreon that can’t glow at all are less likely to be detected in the first place and are therefore actually more effective hunters. (The glow helps, of course, to stun faster prey, but glowing umbreon lack the stealth non-glowing umbreon have.) Additionally, melanin protects the skin from ultraviolet light, synthesizes vitamin D (which in turn is vital to the immune system), and may be associated with viral resistance, all of which in combination allow melanistic pokémon such as your umbreon to be hardier than their normal counterparts. As such, there is some “survival of the fittest” at play here as well.
Finally, as for whether or not I’ve ever seen them in person, I have indeed. While melanism is rare in the wild, domesticated eevee are often bred specifically for those traits due to the fact that besides being ideal traits in the wild, they’re actually more aesthetically pleasing to eevee enthusiasts. However, I must admit I’ve never owned one myself, but I’ve met my fair share of eeveemaniacs with at least one on their team. They’re quite beautiful indeed—so you’re very lucky to own one!