The target is held in its current location
As per the restrained condition, the target's speed is 0, which means it cannot move itself from its current location. In addition, any kind of forced movement is also prevented, per the spell's text. This means that there is no way for the target to move to a different location. If you think in terms of the usual 5-foot grid used for combat encounters, the target is unable to leave its space or be removed from it. Note that even a spell like freedom of movement that prevents the restrained condition cannot release the creature from the chains, because imprisonment specifies that the creature cannot move in addition to the restrained condition. (Although freedom of movement would still prevent the other effects of being restrained for the duration.)
The target is not paralyzed or similar
In plain English, "move" has a number of related meanings. However, in D&D 5e, movement is a defined game term with specific mechanics:
On your turn, you can move a distance up to your speed. You can use as much or as little of your speed as you like on your turn, following the rules here.
Your movement can include jumping, climbing, and swimming. These different modes of movement can be combined with walking, or they can constitute your entire move. However you're moving, you deduct the distance of each part of your move from your speed until it is used up or until you are done moving.
Roughly, "movement" in 5e means translocation: moving from one location to another. This is the kind of motion that is specifically prevented by the chaining variant of imprisonment. It does not include things like turning around in place, making an attack, talking, performing the components of a spell, or any other action that requires moving a muscle but does not cause the creature to move to a new location. A strict RAW reading would allow a chained creature to do all of the above, including casting spells. This means that a chained mage could indeed survive indefinitely, assuming they are equipped with the right spells: mage hand, unseen servant, tiny servant, etc.; as well as perhaps create food and water.
The DM can decide the exact degree of restraint imposed by the spell
Of course, this level of autonomy isn't necessarily consistent with how chaining curses are often portrayed in stories (nor how chains are often used in practice). Frequently, the victim of such a curse has their wrists and ankles bound tightly to the surface they are chained to, making most of the mentioned actions (including somatic spell components) impossible. The Greek god Prometheus is one such example. As the DM, you have the latitude to determine the precise details of what the spell means when it says that "Heavy chains, firmly rooted in the ground, hold the target in place." It could mean that just the target's ankles are chained to the ground, leaving them otherwise unhindered, or it could mean that they are tightly wrapped in chains from head to toe, or anything in between. You could even rule that the decision is left up to the caster according to their intent at the time of casting.
If you are a player planning to learn or use this spell, you should probably talk to your DM and ask how they will handle this ambiguity in the rules. Alternatively, pick another variant of the spell that leaves less room for interpretation. In particular, if you're planning to imprison a mage who might have the ability to cast dispel magic at 9th level, you would want to choose a variant that unambiguously prevents the target from casting spells. In particular, it looks to me like slumber is the only suitable variant for such a high-level mage.
As DM, don't feel obligated to stick to the spell's text
You seem to be asking all of this as a DM designing an NPC in the situation of being magically imprisoned in this way. There is nothing stopping you from designing your own custom magical prison that works in exactly the way you need it to work for your story. You could "implement" this magical prison as an additional variant of the imprisonment spell, but there's no need to do so. You can just declare that the NPC was imprisoned in the specific way you want without having to justify it by referencing a specific spell. Remember that the spells and magic items listed in the rule book are not intended to represent an exhaustive list of all magic that exists in the world -- they are just the spells available for PCs to learn. Encountering magical effects beyond the ability of PCs to replicate is par for the course in D&D.
