Owning a fractional stock is a little like owning a fractional coffee cup.
We think of stock as an investment tool, but that's really more of a useful side effect of the real purpose: indicating ownership or stake in the business, where one share indicates one "vote" in how the company works. The business itself would not recognize a partial share, just like it makes little sense to only cast a fractional vote for an election candidate; you either vote for someone or not. The stock may also be used as a claim on dividend payments, where a fractional share means suddenly trying to pay out fractions of a penny. In a similar way, a fractional part of a coffee cup will have a hard time serving it's real purpose.
But now imagine if coffee cups were made of solid gold. In that case, people might start owning coffee cups that are never used for caffeination, but simply because the value of the material composing the cup can grow over time. Suddenly a fractional part of a coffee cup might make sense, because the material making up the cup is itself also valuable.
In the same way, you never actually "own" a fractional piece of stock. This is a fiction your broker manages for you. If your portfolio has fractional stock, then somewhere your broker owns an actual full share of stock to cover your fractional portion. And if we talk about coffee cups again, instead of cutting a real coffee cup into many pieces the whole cup will rest with one person, who must then keep track of the other partial owners of the cup.
This can work for large brokers with many clients. If 20 clients each own 0.05 of a share, the broker only needs one full share to cover all of them. It's also possible the broker merely took your money and never actually purchased a covering share; instead they will cover it at sale time, possibly even by buying it out themselves. It can be easier and simpler to ensure adequate general cash reserves than a very specific portfolio of stocks who's only purpose is covering many fractional investors. However, they have to be careful about this, since prices can change wildly, and there are rules and regulations governing how certain class firms must manage this risk.
Sometimes allowing fractional shares is not worth the effort at all. One reason is certain stocks can price very high. For example, a single A-share of Berkshire Hathaway currently costs well over $400,000. No broker wants to cover a fractional amount of that stock for one investor, so brokers often need to have limits on this kind of thing. Any broker may also have limits that apply only to smaller accounts.
It sounds like the situation at hand is the broker is perfectly happy to hold fractional amounts of your employer's shares for your account, but doesn't want to deal with this for other stocks. This makes sense, as the employer is likely also a client. The broker will naturally have shares in the employer's stock, and and since all the employees will all have a shares in this one company, the broker need only hold a few shares of the one stock, which it does anyway, and they can cover the fractional amounts for all of the employees.