No, that's not what he said. He got a lot of flak for supposedly saying he invented the Internet.
In reality he never claimed that, he said that when he was a congressman he took initiative in creating the Internet, meaning sponsoring bills that would fund it etc. not actually inventing it.
Robert Kahn and Vint Cerf actually acknowledged that Gore was first politician that saw Internet's potential and helped it becoming a reality.
Also, technically, the actual infrastructure that allows the internet to exist is very much akin to a series of tubes.
Barring satellite internet connections (which are still fairly rare for household usage outside of remote areas), the internet traveled along first phone lines, then coax cables, and now in some places in the US, fiber lines... most of which were and still are underground for all but the proverbial last mile. The internet even crosses seas through giant undersea cables.
If you think of data centers and dns servers as something akin to pumping stations, the analogy of tubes is a fairly decent, simplified metaphor for the actual physical infrastructure of it all. It also allows for a reasonable surface level analogy of water flow through pipes as comparable to bandwidth throughput.
... Though with data cap usage metering, the analogy begins to fall apart at a slightly more technical level, as there is no real... actual 'thing', no resource being 'pumped', beyond encoded electricity itself, which only has a real per unit cost to an ISP in terms of the actual power costs of routing your requests, maybe the maintenance costs of the cable/fiber lines... the former of which is infinitesimally less costly than with water, and said cost is further distributed around the country or world's DNS servers and data server centers.
The only thing that it makes sense to charge for is speeds, and even then, every ISP I've ever used in any location I've ever lived rarely operates at the advertised speed, but technically your contract with the ISP says they don't actually have any real obligation to provide that anyway, and this is apparently legal.
Anyway, only really with the advent of 5g can it really be said that the internet is moving away from being fairly comparable to a vast series of tubes.