A few years ago I learned that some PS2 games and software could go online using a USB modem, but didn't support any other methods like the network adapter. This is because the first network adapter for early PS2s in Japan didn't have a modem, just Ethernet, but dial-up was still common, so an external modem was needed to go online. Then the later versions of the network adapter, which had a modem in the west, left out the modem in Japan, since the USB modems already filled its role.
Several USB modems for PS2 were released in Japan with compatibility varying by game. The oldest modems with the widest support seem to be the "P2GATE" Conexant modem and the "OnlineStation" SUNTAC modem.
These games use direct dial and should work with the same VoIP setup that other direct dial games work with:
Armored Core 2: Another Age
Armored Core 3
Armored Core 3: Silent Line
These games used servers that would have to be reimplemented:
Auto Modellista
Capcom vs. SNK 2: Millionaire Fighting 2001 (cross-platform multiplayer with Dreamcast)
Marvel vs. Capcom 2: New Age of Heroes
Mobile Suit Gundam: Federation vs. Zeon DX (the same person who RE'd the Dreamcast version also did some work on the PS2 version)
I haven't looked into these yet, but I assume they also used servers:
Age of Empires II: The Age of Kings
Eisei Meijin VI
The Seed
Taikyoku Mahjong: Net de Ron!
There are a few web browsers released for PS2 which used the modem, but they also had versions that supported broadband, so using them with modems is just a novelty. Still, they're an easy way to test your modem works with a DreamPi. These are the ones I know of:
EGBrowser
EGBrowser Light For I-O Data Device Inc. (trial version of EGBrowser without HDD and USB keyboard support)
NetFront
NetFront for Delta (should work with both Ethernet and modems, but couldn't dial with my P2GATE modem)
Chou! Tanoshii Internet: Tomodachi no Wa (just reskinned NetFront)
This page has some info. The PS2's library is vast and this feature was relatively obscure so there may be more supported games.