Iceland’s Jon Margeir Sverrisson set a European record of one minute 58.60 seconds to take gold in Eindhoven.
Englishman Hamer, 15, and Welshman Thomas, 19, finished in the same positions at the Commonwealth Games in Glasgow last month.
“This means the world to me – it was a cracking swim,” Hamer told BBC Sport.
Hamer and Thomas’s preparations were disturbed by a fire alarm at their hotel in the middle of the night.
“It’s been a hard few weeks and my head is wrecked, so now I just want to recover,” said Burnley’s Hamer, who clocked a time of 2:00.36.
Nineteen-year-old Thomas, Wales’s only Para-swimming medallist in Glasgow, finished in 2:02.42.
He said: “It’s hard to do a final in the morning so I’m very happy with what I have done here.”
Britain’s Paralympic and world champion Jessica-Jane Applegate lost her world record in being beaten by Russian Valeriia Shabalina in the heats of the women’s S14 200m freestyle.
Shabalina clocked 2:06.50 to better the record of 2:09.88 Applegate set last year, and they will meet again in the final on Friday evening.
Britain’s Matt Walker qualified fastest for the S7 100m freestyle, while compatriot Eleni Papadopoulos was the fastest qualifier in the women’s S10 200m individual medley.
Briton Ellie Simmonds, the Paralympic gold medallist, faces Ukrainian rival Yelyzaveta Mereshko in the S6 100m freestyle final on Friday evening.