Creatures do not fill their spaces, they control them
The PHB tells us that:
A creature's space is the area in feet that it effectively controls in combat, not an expression of its physical dimensions. A typical Medium creature isn't 5 feet wide, for example, but it does control a space that wide. If a Medium hobgoblin stands in a 5‐foot-wide doorway, other creatures can't get through unless the hobgoblin lets them.
It is because of this area of 'control' that friendly creatures are allowed to pass through (typically as difficult terrain), but unfriendly creatures are not, even though the creature does not fill the entire square.
If friendly creatures are permitted to pass through, though, why can they not remain in the area? Because relinquishing 'control' of an area means the creature will fight less effectively, as the quote above continues:
A creature's space also reflects the area it needs to fight effectively.
If a creature is restricted to less space than it needs to fight effectively, its capacity to fight is diminished. This diminished capacity is represented by the rules on squeezing, even if those rules do not successfully address the most-common PC size:
A creature can squeeze through a space that is large enough for a creature one size smaller than it...While squeezing through a space, a creature must spend 1 extra foot for every foot it moves there, and it has disadvantage on attack rolls and Dexterity saving throws. Attack rolls against the creature have advantage while it's in the smaller space.
Although friendly creatures cannot willingly share a space at the end of one of their turns, there are situations in which this might happen involuntarily. Although RAW do not address this possibility, a DM could reasonably consider both allies to be squeezing until one is able to move out of the shared space.
Corpses, as objects, fill smaller spaces than their size category
Typically, when a creature dies - it is no longer considered a creature, but an object. As such, it no longer 'controls' a larger space, but instead merely 'fills' a smaller one. DMs might consider bodies on the floor to be difficult terrain or, as is the case for your table, simply ignore them.
Why is this different from ending a move in a friendly creature's space?
Two friendly creatures forced to share a space can both fight, act, and react - although at diminished capacity. The paladin feigning their death is essentially pretending to be an object, not a creature. To the extent that the DM rules them successful, they should allow the paladin to interact with the world as an object rather than a creature. This follows from Rules 2 and 3: The players describe what they want to do, and The DM narrates the results of the adventurers’ actions. This should bring a whole host of undesirable effects. Some common-sense rules application follow:
The paladin's space is no longer controlled - in the case of your table, the paladin no longer restricts the movement of either allies or enemies.
The paladin is no longer entitled to opportunity attacks when enemies leave his reach, since he is not "constantly watching for a chance to strike an enemy who is fleeing or passing by."
The paladin is at least at disadvantage on some saving throws. He could be considered disadvantaged on Dexterity saves (as if restrained). Alternatively, he could be considered to automatically fail Strength and Dexterity saves (as if paralyzed, stunned, or unconscious). For a DM who has a hard rule about not permitting characters to willingly fail saves, rather than automatically fail, whatever effect provoked the save might instead automatically reveal that the paladin was, in fact, alive.
If an enemy creature suspected that the paladin was faking, he could certainly be attacked at advantage (as if blinded, paralyzed, stunned, or unconscious - although most likely this would be redundant for melee attacks since he is likely prone). More importantly, if the paladin was making no move to defend himself, attacks from within five feet might be considered automatic criticals (as if paralyzed or unconscious).
The paladin, feigning death and perhaps with his eyes closed, would be less observant. He might automatically fail any ability check that required sight (as if blinded) and might not be allowed to use his Passive Perception (as if "not focused on watching for danger").