Tivo Ty Files were an exported version of the database oriented FSID files from an early version of the Tivo operating system. This is purely speculation.. but
T could stand for Tivo
Y could stand for 'Yellow Dog Linux'
Early versions of the Tivo recording box were based on the IBM PowerPC PPC cpu.
A version of Linux compiled for the PowerPC cpu was called 'Yellow Dog Linux' which is where the fancier YUM command later used in RedHat Linux came from. Its was more the Package manager to the package format, as APT was to the debian dpkg file relationships.
Anyhoo.. that the original Tivo operating system probably had some similarities or origins to the Yellow Dog Linux, might explain the inclusion of the letter 'y' in the name.
And since the FSID objects in the storage system were a type of non-file, file which held an entire recording, calling them Ty files .. might make a nice play on words or 'pun' attractive to program writers at the time.