The original PS2 (fat) lacks built-in Ethernet/Wi-Fi. The Network Adapter (disc + PCMCIA card) was sold separately. The Slim (SCPH-70000+) has built-in Ethernet but no Wi-Fi.
Sony never ran a unified "PSN" for PS2. Online was per-game, using publisher-run servers (EA, Sony, Konami). All official servers are shut down.
Verdict: Native PS2 online is dead. Use PCSX2 on PC for online play, or enjoy legendary local co-op.