He claimed serious injury to his reputation.
Can I use damage?
The closure of the factory will cause severe damage to the local economy.
The ship sustained only minor damage.
Can I use injury, instead of damage? Are they synonyms?
If you tell someone about serious injury, the first thing comes is the physical harm to someone.
OALD defines this word and it is mostly used for physical harm
However, injury can happen to someone's feeling (on the same page).
In usage note injury OALD defines being injured where instances are of physically being hurt.
Being a medical professional, I'd like to mention that damage can also happen to human organs (especially internal) but then it could be the result of an injury or harm by bad microorganisms
But again, if you are talking about someone's reputation or injury in the context of non-living thing, use damage to keep ambiguity at bay.
Don't get confuse that damage can be used for physical harm. 'Physical' can be used for non living things like computers and books etc. :)
So, a ship is a non-living thing so it might sustain damage and let the closer cause damage to the economy, not injury. :)
For example: "She was taken to hospital with serious head injuries."
For example: "These chemicals have been found to cause serious environmental damage."
In the case of He claims serious injury to his reputation;
Here, the reputation of a certain person is personified.(why?)
Lets get thorough with the definitions first:
An injury affects a living organism, the likes of you and me
Damage affects a non-living object, like a car
So they can not be used interchangeably.
Lets get back to your word usage:
Is absolutely correct.
Is incorrect and should be replaced with damage.