No. Only creatures are subject to effects that require saving throws.

Spells that do damage based on a saving throw explicitly restrict their effects to creatures, including cantrips. Two examples of many:

Flame Strike:

A vertical column of divine fire roars down from the heavens in a location you specify. Each creature in a 10-foot-radius, 40-foot-high cylinder centered on a point within range must make a Dexterity saving throw. A creature takes 4d6 fire damage and 4d6 radiant damage on a failed save, or half as much damage on a successful one.

Burning Hands:

... Each creature in a 15-foot cone must make a Dexterity saving throw. A creature takes 3d6 fire damage on a failed save, or half as much damage on a successful one.

Ioun stones aren't creatures, so none of these spells (or any other spell that targets creatures) can affect them.

Some such spells also have specific effects against objects; two examples, again of many:

Fireball:

.. ignites flammable objects in the area that aren't being worn or carried.

Shatter:

A nonmagical object that isn’t being worn or carried also takes the damage if it’s in the spell's area.

Neither of these would affect an ioun stone; when in use, it's treated as being worn or carried, and when not in use, it's neither flammable nor nonmagical.

This GM would rule that if two or more otherwise identical ioun stones grant bonuses on different rolls or to different values, a creature can gain the benefit of each otherwise identical ioun stone.

For example, this GM would have no difficulty ruling that a creature that sends spinning both an ioun stone (dusty rose prism) and an ioun stone (cracked dusty rose prism) then benefits from both the first stone's +1 insight bonus to AC and the second stone's +1 competence bonus on initiative checks.

Likewise, this GM would rule that the same creature can benefit both from an ioun stone (cracked pale green prism) that grants a +1 competence bonus on attack rolls and from an ioun stone (cracked pale green prism) that grants a +1 competence bonus on saving throws. Bonuses on or to different things that are from different sources generally stack, and in this case the sources are two different ioun stones, each having a different effect.

However, a stricter GM may want to apply the rules for Combining Magic Effects on Stacking Effects on Same Effect with Differing Results (noting, however, that such rules are usually applied in the context of spell effects) and declare the two identical ioun stones the same effect (for example, two ioun stones (cracked pale green prisms)) with differing results (for example, a bonus on attack rolls and a bonus on saving throws). Such a ruling would mean the creature benefits only from the last-sent-spinning otherwise identical ioun stone. This GM does not endorse such a ruling, but such a ruling wouldn't cause this player to abandon a campaign.

(Also, because the text on d20PFSRD is slightly incomplete, "For stones that have multiple possible effects…, the exact effect is determined by the item crafter when it is created and cannot be changed thereafter" (Seekers of Secrets 46). Thus, for example, it's not like a creature that finds an ioun stone (cracked pale green prism) and sends the stone spinning around its head itself picks the bonus; instead, the stone's creator picked the bonus when the stone was created.)

In Dungeons and Dragons

I found nothing like your DM's recollection of the limit on ioun stones in the Dungeon Master's Guide for AD&D, 2e, 3e, or 3.5e. Also silent is the Encyclopedia Magica for AD&D 2E (the game's largest magic item collection so far) in its entry on Ioun Stones (613-8). (Right after Internal Conjuration Engine but before Ipsissimo's Black Goose.)

Further, Hargenrader's Dragon #174 (Oct. 1991) Bazaar of the Bizarre column "Ioun Stones: Where Do You Go If You Want Some More?" (90-4) references the Polyhedron #15 Dispel Confusion column that seems to indicate ioun stones can increase an AD&D 2e character's ability scores not only above their racial maxima but also even above 18! Despite mentioning this unusual effect, Hargenrader's article doesn't discuss—rather surprisingly, in retrospect—the possibility of a creature employing several identical stones. (However, given the frequently punitive nature of old school D&D, this reader expects any attempt to send multiple identical stones spinning to yield explosions.)

Finally, the Encyclopedia Magica lists ioun stones as possibly either burnt out, normal, supercharged, or cursed. Inferior stones—cracked, flawed, and scorched—and resonating stones are, presumably, a Pathfinder invention so any information about a creature specifically being unable to use simultaneously both a typical stone and an otherwise identical inferior stone would have to buried even deeper within one of Pathfinder's multitudinous supplements.

Note: I also looked at Pelgrane Press's Primer of Practical Magic (2004). The book is billed as "Spells and artifacts from the Dying Earth, made available for use in [the] d20" System, and includes fifty or so additional ioun stones (some underpriced, some unbalanced, and others both—introduce them cautiously to a campaign!), but that text, too, says nothing about multiples or identicals.