By a fine reading, the Accursed Specter rises before it can be Soul Caged
This gets into a rather fine reading of the RAW (and assumes that "spirit" and "soul" are used more or less as synonyms in 5e), but I have to conclude that the Accursed Specter feature has the priority. Here's why:
If (contrary to fact) the Accursed Specter feature description was written as: "After you slay a creature and its spirit rises from its corpse, you can cause that spirit to become a specter" then it would mean that the specter is created after the departure of the spirit from the body, and then it might lose out to soul cage.
However, that's not how Accursed Specter is described. Rather than saying that the spirit first rises and then becomes a specter -- it says it rises as a specter. And it says this happens not after you slay the creature, but when you slay the creature. So when using this feature, as the spirit departs the body (which is when that creature is truly slain), it is already a specter, because it rises from the corpse "as a specter."
Next, we can see that this is going to have occurred before soul cage would take effect. We know that the caster of soul cage uses their reaction to cast the spell, and as for the timing of reactions, XGtE says:
If you’re unsure when a reaction occurs in relation to its trigger,
here’s the rule: the reaction happens after its trigger completes,
unless the description of the reaction explicitly says otherwise.
And the soul cage spell description says:
This spell snatches the soul of a humanoid as it dies and traps it inside the tiny cage you use for the material component.
This seems to portray the spell as "snatching" the soul immediately when it exits the corpse, presumably before it can pass on to wherever else it would go in the afterlife.
But if the Accursed Specter feature has been applied when the creature dies so that its soul rises as a specter already, then it's too late for soul cage. When the soul escapes the corpse and starts to rise, it is already transformed into an undead creature.
The spirit affected by Accursed Specter never has a moment in time when it is "up for grabs" because it rises as a specter. Soul cage can "snatch" a departing spirit, but it cannot capture one that is no longer escaping to anywhere, but rather has already begun its undead existence as a new type of creature.