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Originally Posted by Selmo

"and was never heard again in that age of this world."

If "that age" refered to the Third Age then it would have been written as "Age", not "age".
Tolkien would have been fussy about details like that.
.

Not neccessarily.

So passed the first victim of the malice of the masterless Ring: Isildur, second King of all the Dúnedain, lord of Arnor and Gondor, and in that age of
the World the last. From the Disaster of the Gladden Fields

Here, the 'age' referred to is clearly the Third Age.

Also,

For, my lords, it
may well prove that we ourselves shall perish utterly in a black battle far from the living lands; so that even if Barad-dûr be thrown down, we shall not live to see a new age. But this, I deem, is our duty. And better so than to perish nonetheless - as we surely shall, if we sit here - and know as we die that no new age shall be."From the Last Debate

and after the ending of the Third Age of the world into the new age it preserved the memory and the glory of the years that were gone. From the Steward and the King

and though all things may seem changed, as if an age of the world had gone by, yet to the trees and the grass it is less than a year since you set out.From the Steward and the King

You will be the Mayor, of course, as long as you want to be, and the most famous gardener in history; and you will read things out of the Red Book, and keep alive the memory of the age that is gone. so that people will remember the Great Danger and so love their beloved land all the more. From the Gray Havens

There are other examples where Tolkien uses 'age' to refer to an Age of the World. So I don't think he was very fussy about that.