The ship is not a creature and thus automatically takes the damage
The DMG Chapter 8: Running the Game states (emphasis mine):
For the purpose of these rules, an object is a discrete, inanimate item like a window, door, sword, book, table, chair, or stone, not a building or a vehicle that is composed of many other objects.
Ships have the type vehicle. So they are not an object per se, but they are composed of objects. Thus we can say they are a composite object.
A nonmagical object that isn't being worn or carried also takes the damage if it's in the spell's area.
This ship is a non-magical object that isn't being worn or carried, and thus takes the damage automatically (or at least it's components do).
The Ghosts of Saltmarsh has specific rules on each of the components, and how they take damage, but they do take damage.
A ship is composed of different components, each of which comprises multiple objects:
Hull. A ship’s hull is its basic frame, on which the other components are mounted.
A component has an Armor Class. Its AC reflects the materials used to construct it and any defensive plating used to augment its toughness.
A ship component is destroyed and becomes unusable when it drops to 0 hit points. A ship is wrecked if its hull is destroyed.
A ship doesn’t have Hit Dice.
If a ship component has a damage threshold, that threshold appears after its hit points. A component has immunity to all damage unless it takes an amount of damage that equals or exceeds its threshold, in which case it takes damage as normal. [...]
So, the components in range of shatter will automatically be affected by the damage, but if they have a damage threshold the component is immune to any damage unless it exceeds the damage threshold.
Normally objects don't make saving throws. The general rules on who or what makes Saving Throws are available in the Players Handbook section on Saving Throws:
A saving throw — also called a save — represents an attempt to resist a spell, a trap, a poison, a disease, or a similar threat. You don’t normally decide to make a saving throw; you are forced to make one because your character or monster is at risk of harm.
Ghosts of Saltmarsh introduces a number of new and specific rules around ships.
The ability scores on the ships stat block have new and specific meanings outlined in the Ghosts of Saltmarsh:
The Strength of a ship expresses its size and weight. Dexterity represents a ship’s ease of handling. A ship’s Constitution covers its durability and the quality of its construction. Ships usually have a score of 0 in Intelligence, Wisdom, and Charisma.
If a ship has a 0 in a score, it automatically fails any ability check or saving throw that uses that score.
This passage also establishes that there are situations where ships will need to make ability checks and saving throws. This is a specific exception to the general rules, quoted earlier, that creatures make saving throws.
In particular Ghosts of Saltmarsh introduces new rules for Crashing a Ship in which the ship needs to make a Constitution saving throw.
Crashing a Ship
When a ship crashes, it must immediately make a DC 10 Constitution saving throw. [...]
Vessel based saving throws are also required when fighting against currents:
[...] Vessels and creatures fighting against a current must make a Constitution saving throw at the end of every hour traveled, with a bonus equal to the crew’s quality (if they have crew). [...]
or when navigating sandbars:
[...] When a ship passes over a submerged sandbar, the sandbar counts as difficult terrain, and the ship must succeed on a Dexterity saving throw. [...]
There are also a number of specific upgrades and magic items ships can have that can force new saving throws, for example:
This weapon creates spectral chains that trail after its shots, restraining enemy vessels in their grasp. When this weapon hits a ship, that vehicle must succeed on a DC 14 Strength saving throw or it can’t move away from the attacker. [...]
Red Dragon Figurehead
[...] Each creature and object in the cone must make a DC 12 Dexterity saving throw, taking 21 (6d6) fire damage on a failed save, or half as much damage on a successful one. [...]