During this scene in Game of Thrones where the Kingsguard go to kidnap Arya and Syrio fights them, Syrio uses his wooden practice sword against their real metal swords.
Syrio had several opportunities to replace his wooden sword with a metal one, most notably 40 seconds into the scene when he beats down one knight with a blow to the head and has ample time to pick up that knight's sword.
I doubt it's because the knights' swords were incompatible with his fighting style. They almost certainly would've been preferable over a broken wooden practice sword. Also, as a former First Sword of Bravos, Syrio would've been experienced with many fighting styles and weapons - he of all people should know that combat can happen at any time and doesn't always give you your weapon of choice.
If Syrio chose his wooden sword over a metal one not for practical reasons, did he do so out of pride? Pride bringing about a man's fall seems to be a common theme in Game of Thrones. The fact that he preferred his broken wooden sword over Arya's also supports this.