I know that by lore, giving an unseen servant some semblance of separateness from the caster is possible in that a published adventure contains at least one example of this occurring:

In Waterdeep: Dungeon of the Mad Mage, level 2, area 11b. Master of the Unseen (p. 35), Midna Tauberth, a priestess of Shar, is waited on by 9 living unseen servants left behind by one of Halaster's apprentices. It states:

"The unseen servants obey Midna’s commands because she reminds them of their creator. They follow Midna and can leave the room to fulfill her wishes. Midna has no way to distinguish one servant from another, meaning that when she utters a command, all the servants follow that command. She can’t issue orders to a single servant without the others performing the same task."

How can a character make these using the normal unseen servant spell and presumably something else?

(Simply saying that there isn't anything in the spell description that says whether this is possible, or simply referring to the "Spells do what they say they do" rule, is a little obvious and would be unappreciated.)

I need an answer that explains how the apprentice made these by rules or by lore.

There is no official answer

As far as I am aware there is no official 5e rule that allows you to create permanent living unseen servants. This is likely one of those cases where NPCs have things that there is no RAW way to obtain.

Previous editions had a spell called permanency which would allow you to make the effects of certain spells permanent. However unseen servant was not one of the spells that this would work on.

Unfortunately the answer is "up to the DM". I know this is unsatisfying and maybe someone with better researching skills than me will find a better one, but it is the only one covered by 5e RAW.

RAW, there is no documented way to do this

This comes under the category of "DM prerogative". Many adventures will use magical places or situations for which there are no rules by which a character can replicate them (flying castles, extra-dimensional huts that walk about on chicken legs and so on).

Such things can always exist of course if a DM (or the adventure writer) says so. But that doesn't always mean there has to exist a set of rules that say "here's how you do it".

A DM is free to whip up whatever his or her imagination desires!

“A wizard did it”

You’ve probably heard the phrase before to explain with a hand-wave why something weird exists, like bulettes or owlbeards: “a wizard did it.” There are things in D&D that were made by human Dungeon Masters for convenience of creating fun adventures, not via rules for players.

Like owlbears were (allegedly) created by a mad wizard, so is a lot of Undermountain.

Halaster’s apprentices are mad and powerful inventors

Halaster’s apprentices are older than Waterdeep. The city was built on the ruins of their original wizards’ tower. Today, those who still live, are ridiculously powerful archmages. Halaster himself, if he still lives (the jury is still out), is nearly a god-like force of nature in his own home.

The entire existence of Undermountain was originally to give Halaster and his apprentices a place to learn secrets of magic and develop new spells and things in private. The place is packed with bizarre, unique creations that have not been shared with wizards outside its halls.

His apprentices guard the secrets of their discoveries and creations jealously (first and foremost from each other — only secondarily do they care about the world above), and nobody knows how they do what they do.

In a less lore-focused way, Halaster and his apprentices were created to be plot devices, which could be used to explain anything non-standard that exists in Undermountain.

Every spell was invented by some wizard once

Earlier editions of D&D had spell research rules, and those are part of the lore of even 5th edition. For example, Tenser’s Disc was invented by Tenser (a PC played by Gary Gygax’s son Ernest in Gary's campaign). The spell wasn’t available until Tenser researched it and shared it with other wizards, so that knowledge of the spell spread throughout Greyhawk and, before long, across the Multiverse.

Halaster and his apprentices have invented many spells, rituals, and partly-magical inventions. You will find many unique magics under Waterdeep, but won’t necessarily learn the secrets behind them easily, if ever. Those secrets are buried deep, and they don’t like to share.

There is probably a ritual to create more of these living unseen servants, but it’s likely hidden away in the carefully guarded spellbook in the carefully guarded secret lair of a mad and dangerous archmage somewhere deep in the lower levels of Undermountain.

To read more

The original lore on Halaster and his apprentices is first established in their first published appearance, the AD&D 2nd edition megadungeon boxed set The Ruins of Undermountain. The price of the PDF is well worth it just for the lore, but it’s an excellent playground for PCs of all levels as well. (Converting it to 5e is likely a pain though, and I haven’t tried yet so can’t recommend either way.)

While it is true that there is no such spell as living unseen servant described the the published materials, that does not mean it is flat out not possible. Why, the world full of possible things that were not possible at some point.

As always, it depends on the DM. If the DM would allow such a thing, it might be possible. If you are asking as a player, talk to your DM. If you're asking as a DM, you have to ask yourself if you would want such a thing in your campaign.

After all, every spell was presumably invented by someone. You could even argue that either a) the world contains infinite examples beyond the power of even the most powerful player character, which personally I think is pretty lame or b) somehow, some way, with enough effort, what the PCs see, they could aspire to do.

Doesn't mean it's easy.

In Xanathar's Guide to Everything, a downtime activity of research is described (p. 132-133). A character could attempt to research an answer, or an answer, or even just information or a hint.

The time and cost of research is measured in weeks and hundreds of gold for even a few pieces of lore, and typically requires access to a library or a sage.

Xanathar's also describes crafting a magic item (p. 128-129). While apparently intended for creating items described in the published materials, it is not such a stretch that a DM might allow a character to craft a magic item that is different from the ones described in the books.

The description for crafting magic items includes:

Creating a magic item requires more than just time, effort, and materials. It is a long-term process that involves one or more adventures to track down rare materials and the lore needed to create the item.

It is not such a stretch that a character could research a new spell.

Such an endeavor might be similar, or a combination of, lore research or crafting a magic item. Of course, the DM could choose to make such an effort easier or harder than that. A wizard can create an unseen servant, create a homunculus, create undead, capture the souls of others. Surely somewhere in there creating a living unseen servant is not completely out of the same league.

Making the effect permanent, that does seem like it's much harder. Maybe that will take more research.

And of course, there is wish. A wizard could attempt to wish for a living unseen servant, or wish for a servant to become unseen, or wish an existing living unseen servant to acknowledge the wizard as master, or many other permutations.

The are other possibilities, which, are admittedly, a bit of a stretch, but maybe not completely. Clerics can call for divine intervention. Sorcerers have powerful patrons. In either case, perhaps a character could gain a permanent living unseen servant as part of a (no doubt bad) bargain of some sort.

As another answer said, this answer may be a bit unsatisfying, and is definitely pushing at the boundaries of RAW, but I don't think it's out of the bounds of RAI. The game is intended to be extensible and customizable, and this is definitely one of those cases.