Multiplying damage is a known, defined thing in 3.5e:
Sometimes you multiply damage by some factor, such as on a critical hit. Roll the damage (with all modifiers) multiple times and total the results. Note: When you multiply damage more than once, each multiplier works off the original, unmultiplied damage.
Exception: Extra damage dice over and above a weapon’s normal damage are never multiplied.
There are a number of examples of multiplying damage, including the core examples of critical hits (which repeats the above rules, and then refers to it as being “doubled” or otherwise “multiplied”) and lances in a mounted charge (“A lance deals double damage when used from the back of a charging mount”). There are also numerous non-core examples like the ruby and diamond nightmare blade Diamond Mind maneuvers from Tome of Battle, the decisive strike alternate monk class feature Player’s Handbook II, and... the manyfang dagger.
For extra confirmation, we have the \$\times5\$ multiplier on a critical hit: that is consistent with a weapon that has a \$20\$/\$\times2\$ critical property, but comes with a \$\times4\$ multiplier default: then, when you roll a critical hit, you perform the usual odd multiplier addition:
\times4 \times2 &= \\
\times1 + (\times4-1) + (\times2-1) &= \\
\times(1 + 3 + 1) &= \times5
If the manyfang dagger referred only to the base damage die when it was talking about quadrupling, it would be very weird to then refer to the entire damage expression when quintupling. If it referred only to the base damage die, I would instead expect the 4d4 to be doubled, not everything to be quintupled.
And, of course, if they just meant 4d4, why go through this whole rigmarole instead of just saying it has a 4d4 base damage die?
So no, “quadruple damage” is not a fancy way of saying 4d4. It is saying that you “Roll the damage (with all modifiers [excepting extra damage dice over and above a weapon’s normal damage]) multiple times and total the results,” and that makes the manyfang dagger preposterous. So you should probably just ban it, and should give Serpent Kingdoms in general a pretty critical glare. Certainly don’t let anyone shapechange into a sarrukh, is what I’m saying. (But seriously, don’t stop there; ability rip and venomfire also spring to mind immediately as theoretical optimization stars. There very well may be more.)