The game is always whatever the DM wants it to be, so the answer is "however the DM wants this to happen".
Dale M's answer covers the most important criteria here: the rules are not detailed enough to describe this, so there are no rules-based contradictions (or supports) to deal with, and all elements of a D&D game are the DM's purview, so rules-based contradictions would not be a constraint anyways.
Since you've made clear that the nature of this particular D&D game is such that the situation doesn't need to "make sense" in any way, that's your answer: an area can be a lair for as many creatures as your DM feels are appropriate, and the lairs are established through whatever means your DM feels are appropriate.
The domains of liches and unicorns are described very differently in the MM:
A lich often haunts the abode it favored in life, such as a lonely tower, a haunted ruin, or an academy of black magic. Alternatively, some liches construct secret tombs filled with powerful guardians and traps.
Everything about a lich’s lair reflects its keen mind and wicked cunning, including the magic and mundane traps that secure it. Undead, constructs, and bound demons lurk in shadowy recesses, emerging to destroy those who dare to disturb the lich’s work.
A unicorn’s lair might be an ancient ruin overgrown with vines, a misty clearing surrounded by mighty oaks, a flower-covered hilltop alive with butterflies, or some other serene woodland location.
There's no reason these couldn't overlap: a secret tomb might lie beneath a serene forest, perhaps. And even these published suggestions are qualified by words like often and might. This particular pairing might be easier to assemble than some others, as (per the MM) liches have lair actions but no regional effects, while unicorns have regional effects but no lair actions.
With other creature combinations I might be more concerned about an area being fundamentally suffused with both evil and good energy due to the presence of two magically powerful creatures, but this is not a mechanical element of the game and the fact that only the unicorn really has this property here (plus, again, the unlimited DM discretion) makes it even less relevant.
There are a few elements to bear in mind in how a DM might rule on these matters, which are relevant to the question and game mechanics but don't constitute a rules-based, definitive result on whether or not this would be allowed or prohibited:
- There aren't any official, recorded instances of an area being a lair for multiple creatures (that I am aware of; if anyone knows otherwise please leave a comment and I'll update). Given how valuable lair actions and regional effects are, if this plan worked as described in the question we might expect to have seen at least some creatures taking advantage of stacking these effects.
The rules offer specific systems for various benefits to be available to players. Creating a lair on demand to enjoy regional effects is not one of these. It's true that the rules do not explicitly prohibit it (and, as above, rules are irrelevant to a DM's discretion in the first place), but they also don't prohibit a PC developing a cantrip that kills or destroys any target instantly without any chance of failure or misadventure.
Since we're going beyond the explicit boundaries of the rules to enact new mechanics, this is fundamentally a homebrew situation, and homebrew cannot be defined by existing rules.
- Prepare for cheese. Regional benefits and lair actions can be very valuable, and so if this is an option for your group then the natural extension is that your group will do this sort of thing often. Both in terms of sending the unicorn into areas to en-lair them, and also in the form of binding additional creatures to get more lair actions and regional effects out of the strategy.
There is no inherent reason for an antagonist to allow something else to have free rein in its lair. Some might, but the nature of a lair is such that its owner would probably discourage loiterers and long-term squatters. You can say that this is a hack-and-slash game, but that doesn't really change anything-- the lich's response may well be to hack and slash at the unicorn for barging into its home, just as it would when the PCs themselves enter during another phase of what is the same errand.
A DM does not need justification for setting up a scenario beyond a belief that it will make the game fun and interesting, but if that's the justification for nothing to matter but the outcome of the setup then a bunch of ad hoc rules around lair creation are overdetermined.
The effort of creating a lair is unlikely to be trivial. After all, simply having the unicorn nearby isn't enough to produce these effects-- its magic and/or mehcanical preparations need to be present throughout the area for the effects to happen.
The players have likely invested a good deal of time and effort into enslaving the unicorn, and as a result probably expect to get some real utility out of it. That utility does not need to be infinite: if they have to go without the unicorn's aid for six months while it establishes itself in the lich's lair, that's a meaningful investment of the party's hard-earned resources. If it takes twenty minutes instead it's a different story.
If the level of investment is not important, homebrewing some other method to provide the regional effect benefits might be easier to do. Just declaring that the unicorn has established a lair for itself is even easier than that, and needs no fluff (which seems to be not relevant here anyhow).