The site linked is using FLITE or Festival Light, an optimized open source reimplementation of the Festival speech framework. Development goals where fast(er) generation, (relative) small footprint and low power consumption so it can be used on upcoming PDAs of the early 2000s.
Festival is developed since 1995 at the University of Edinburgh, while FLITE development happens at Carnegie Mellon University since 1999.
Being rather recent (post 2000) I wouldn't really consider this software retro. Though, the underlying synthesis model dates back to the 1970s.
If your memory is really about something classic, maybe you remember some system using an SSI 263 (Votrax SC-02) chip? There have quite a lot boards using that chip all thru the 80s and early 90s. TRS-80, PET and Apple II all the way to PCs got their speech boards.
Don't get fooled by the many less than great examples on Youtube, as they usually only show the basic capabilities, where the majority of TTS is done in with the chiptables, which naturally only give a rather robotic voice. There has been better adjusted software available fine-tuning the output with results much more like what the FLITE examples show.