Not really, so long as you avoid double advancement
This doesn’t really introduce much that’s actually new, and existing options that are similar don’t really break anything—they might allow the bard to access game-breaking stuff they couldn’t easily leverage before, but other classes could so at worst it just lets bards get in on that. (And the real answer is to ban the game-breaking thing, of course.)
Official options can already get you close
There are a number of ways within the official rules to—arguably—achieve something quite similar to this anyway. Note that there are potential “issues” with each of these—sha’ir is Dragon content, Southern Magician gets into all kinds of RAW arguments, and divine bard is an optional variant—but nonetheless, a lot of games and a lot of online discussion includes these things and so already allows something quite similar to what you’re envisioning.
Notably, however, none of these things allow a single spell to be cast as both arcane and divine simultaneously. That would be the one thing your ruling would introduce as a new feature.
Sha’ir natively cast arcane spells and divine spells
Dragon Compendium is, as the name suggests, a compilation of material from Dragon magazine articles. Those are advertised as official D&D material. And one of the things in Dragon Compendium (or from some Dragon article) is the sha’ir, a weird spellcasting class that uses a familiar that’s a mini-genie, who goes and retrieves spells from the Elemental Planes. Yes, it’s odd. Anyway, one of the myriad ways in which it’s odd is, the sha’ir can have its gen familiar retrieve arcane spells (from the sor/wiz list) or divine spells (from one of several domains). So that’s already a spellcaster that’s both arcane and divine. Each spell is either arcane or divine, so it’s not quite the same thing, but it’s pretty close.
Southern Magician arguably lets a bard do the same
The feat Southern Magician (Races of Faerûn) already does most of what this ruling would accomplish, for any arcane spellcaster. Southern Magician allows an arcanist to treat a spell as divine—instead of arcane—if they like. Like with the sha’ir, the spell isn’t both arcane and divine at the same time, but it can be either at the spellcaster’s discretion. How far it goes is a matter of intense debate—particularly whether an arcane spellcasting class counts as a “divine spellcasting class” for prestige classes if you have this feat. Bards can take Southern Magician without trouble, though it takes more levels of bard to meet the 2nd-level spells requirement.
Unearthed Arcana has a “divine bard” variant
For that matter, Unearthed Arcana has a variant bard that is divine. It’s not both arcane and divine, but it does mean that, for example, that archivist could theoretically have accessed bard spells through the divine bard. It’s a variant, but since it’s in the SRD, the divine bard is pretty widely allowed.
Double-progressing bard has to be banned
No single prestige class level can be allowed to progress bard spellcasting twice. For instance, you cannot allow anyone to take a level of mystic theurge, and then choose “bard” for their arcane spellcasting class and then “bard” again for their divine spellcasting class. Even with bard’s weaker spellcasting, that rapidly becomes absurd.
Sha’ir already ran into this issue, and some argue—very controversially argue—that Southern Magician can do it too. Whether it’s sha’ir, Southern Magician, or your bard, it cannot be allowed. Even with bard’s weaker spellcasting, a 4th-level bard/6th-level mystic theurge—allowed to double-progress bard—casts 6th-level spells at ECL 10th (one level before a cleric, druid, or wizard gets 6th-level spells). It’s notable that it only happens at that one level—those classes catch up immediately after that level and then progress beyond what the bard can do even with double-progression—but this was also the most simplistic, naive entry (early entry tricks could easily do a 1st-level bard/3rd-level mystic theurge, surpassing full-casters at ECL 4th and maintaining that advantage until ECL 11th), plus really you just do not want this precedent.
Sublime chord needs a ruling
Sublime chord (Complete Mage) is a prestige class that gets its own spellcasting of 4th-level through 9th-level spells from the bard and sor/wiz lists. Basically, it’s a fancy bard extension that allows a more sorcerous bard with higher level spells. Since they’re kinda-sorta bards, you need to rule on whether their spells gain the same benefit as bards.
I can’t imagine any possible ruling here that’s going to cause any additional issues, though. (Obviously, if you choose to allow them to have the arcane-and-divine spells, it makes the mystic theurge issue that much more potent, but it already was crucial so that’s a bit moot.)
Otherwise, Divine Metamagic is the only real concern
Of all the things that specify divine spells, Divine Metamagic is the only really big problem. But it’s a really big problem for clerics, and for bards, who don’t have as strong spellcasting or native turn undead, it wouldn’t be as large a problem.
(There’s also even a Metamagic Song feat that’s similar for bards, but it has a crucial limitation that Divine Metamagic lacks: it can’t allow you to cast the spell if the metamagic’s usual adjustment would increase the spell’s level above that which you could actually cast. There’s actually ways around that, too, but it’s so many hoops to jump through that just “get turn undead” would undoubtedly be easier.)
Arcane-and-Divine doesn’t really matter much
The thing about arcane spells is, being “arcane” is usually pretty much all downside. The advantage of arcane spells is that the spells themselves are usually better—which is why it’s worth ditching armor, why clerics and druids get major class features in addition to their spells and sorcerers and wizards do not, and so on. Divine Metamagic is an astoundingly powerful feat—and there’s just no similar thing on the arcane side. Even powerful “arcane-only” prestige classes—like incantatrix—still work if you use Southern Magician on your arcane spells to turn them divine, and then use Divine Metamagic on them.
So the one new thing that this ruling actually introduces, I’m not sure it ever ends up mattering.