This seems to be an oversight in the TV Show
Note: I am not a sword smith or a metallurgist.
The only thing that would seem to set a freshly forged sword apart from an older sword is the sharpness of the blade, it's strength, as well as any nooks in the steel from clashing with other swords, however as seen from the quote in the question and the one below: Valyrian steel was known for it's strength and ability to hold an edge:
Those arts are now lost, though the smiths of Qohor claim to still know magics for reworking Valyrian steel without losing its strength or unsurpassed ability to hold an edge.
The World of Ice and Fire - Ancient History: Valyria's Children
Jaime knowing that the swords were freshly forged because they looked "new" doesn't quite hold up to the characteristics of the swords, however, we know that Jaime could've deduced that the swords must've been freshly forged from his knowledge of Valyrian Steel and the ones present in Westeros.
Jaime would've been aware of the tale of King Tommen II Lannister
After the Doom of Valyria, King Tommen II Lannister took his great fleet to Valyria in attempt to plunder the Freehold after it's destruction. With him he took Brightroar, which perished in the Doom alongside King Tommen and his fleet.
The sword Brightroar came into the possession of the Lannister kings in the century before the Doom, and it is said that the weight of gold they paid for it would have been enough to raise an army. But it was lost little more than a century later, when Tommen II carried it with him when he sailed with his great fleet to ruined Valyria, with the intention of plundering the wealth and sorcery he was sure still remained. The fleet never returned, nor Tommen, nor Brightroar.
The World of Ice and Fire - The Westerlands
8 years prior to the events of A Storm of Swords, Gerion Lannister went to the Doom of Valyria in the hopes of finding Brightroar after several failed attempts by Tywin to purchase the Valyrian Steel swords of lesser houses.
At long last, Father? Valyrian steel blades were scarce and costly, yet thousands remained in the world, perhaps two hundred in the Seven Kingdoms alone. It had always irked his father that none belonged to House Lannister. The old Kings of the Rock had owned such a weapon, but the greatsword Brightroar had been lost when the second King Tommen carried it back to Valyria on his fool's quest. He had never returned; nor had Uncle Gery, the youngest and most reckless of his father's brothers, who had gone seeking after the lost sword some eight years past.
Thrice at least Lord Tywin had offered to buy Valyrian longswords from impoverished lesser houses, but his advances had always been firmly rebuffed. The little lordlings would gladly part with their daughters should a Lannister come asking, but they cherished their old family swords.
A Storm of Swords - Tyrion IV
From the above it seems rather clear that a large red flag would've appeared when Tywin had appeared with a Valyrian steel sword, especially one of such strange colour compared to most Valyrian steel swords:
"The colors are strange," he commented as he turned the blade in the sunlight. Most Valyrian steel was a grey so dark it looked almost black, as was true here as well. But blended into the folds was a red as deep as the grey. The two colors lapped over one another without ever touching, each ripple distinct, like waves of night and blood upon some steely shore.
A Storm of Swords - Tyrion IV
We can guess that as the favourite son of Tywin, and master swordsman, Jaime would likely have been taught in the lore of the available Valyrian steel swords, what they looked like and their magical characteristics. Upon being presented with the blade Jaime would likely immediately have been aware that his was a new sword, and may therefore have deduced that it was freshly forged.