The rules are woefully under-defined for this question. Far too much inference and context is necessary to reach any conclusions. Often the best rules we have available are only distantly related to what we actually want to know, and we are forced to extrapolate from imperfect comparisons.
Despite all of that, I am confident that there are some firm conclusions we can draw. I am going to treat this answer as a kind of informal “proof,” where I show contradictions in the rules with certain ideas that one might have about this combination.
Combining Magical Effects
Spells or magical effects usually work as described, no matter how many other spells or magical effects happen to be operating in the same area or on the same recipient. Except in special cases, a spell doesn’t affect the way another spell operates. Whenever a spell has a specifi c effect on other spells, that spell’s description explains that effect. Several other general rules apply when spells or magical effects operate in the same place.
One Effect Makes Another Irrelevant
Sometimes, one spell can render a later spell irrelevant. Both spells are still active, but one has rendered the other useless in some fashion.
(Player’s Handbook pg. 171-172, Rules Compendium pg. 137, SRD)
Wild shape is not a “spell,” but the headers here emphasize that this describes “magic effects,” and truly, non-spell magic effects don’t have any other rules found anywhere else.
The emphasis is on one magic effect not interfering with another magic effect, and if they truly must interact, the interaction is limited as much as possible. To wit, if your wild shape effect means that some aspect of bite of the were-X doesn’t apply—no hands to turn into claws—that doesn’t cause the spell to fail or be impossible to cast. It is “still active,” and should apply as much of its effect as possible—meaning that if you wild shape into a snake, bite of the were-X can still affect your face and give you a bite, even though you don’t get the claws. If you end your wild shape before bite of the were-X runs out, then it is still active, and you return your normal form with the effects of bite of the were-X applied, claws included. To claim that the spell fails entirely and none of its effects apply is wrong.
Natural weapons generally require a limb or other extremity with which to use them. The rules don’t discuss this requirement explicitly or in great detail, unfortunately. Instead, it’s the conclusion we must come to based on some descriptions and the examples given in creatures’ statblocks. This comes up most often when someone uses a clawed hand to attack with a manufactured weapon—they can’t also use the claw on that hand in the same attack sequence—but nothing actually says that, we just see it in monsters’ statblocks. It also matters a great deal when transmutation effects are applied that give multiple natural weapons. For instance, if you cast two different spells that say you get a pair of claws, but you only have two arms, you can’t make four claw attacks—you just choose the better pair. So if you wild shape into something with claws, and then cast bite of the were-X that grants claws, you’ll have to choose which set of claws you’re actually using. This is also consistent with the same combining magical effects section referenced above, which also includes a section on “same effect more than once in different strengths” and says “only the best one applies.” And we have examples of effects that explicitly overcome this limitation, such as the girallon arms soulmeld from Magic of Incarnum that explicitly allows you to use four claw attacks with only two arms (by giving you two magical arms for the two other claws). Therefore it would be wrong to claim that wild shape and bite of the were-X could combine to give more claw attacks than one has hands, or multiple bite attacks when one has only one head, and so on.
Moreover, the rules for magic items—the only rules we have that discuss different body forms in much detail—specify that most creatures have the same slots as humanoids. Quadrupeds use their forelegs as arms and hands (Magic Item Compendium pg. 219, Rules Compendium pg. 84), and so on. There are some body shapes that lack slots entirely, but the rules generally try to make things work the same for everyone. Therefore, getting hung up on words like “face” or “hands” when a creature has similar extremities is inconsistent with the rules we do have. It is thus difficult to say it’s “wrong,” but it is at least inconsistent with the general thrust of the rules to distinguish between, say, “hands” and a bear’s paw.
Addressing specific points
Obviously, the idea of these spells is a "lycanthropic" transformation of the caster. Since only humanoids and giants can be lycanthropes, the reference to hands, mouth and face makes sense. [...]
although it's not a polymorph spell, the wording of the "Bite of the Were-X" spells suggest that the benefits you gain arise from becoming a half-animal creature. But would the benefit of being a, let's say, half-bear not overlap with that of being a bear?
This isn’t obvious at all—for all we know, the bite of the were-X spells are named that because of perceived similarities between the effects of the spell and the effects of lycanthropy, but that the two have absolutely nothing to do with one another and the similarity is coincidence. Nothing in the spell descriptions says anything like “this magic gives you a form of temporary lycanthropy,” or anything like that. It’s not at all clear how much we should or should not read into the names of the spells. After all, names are things that people apply to the spells, not inherent properties that they come with—I mean, presumably. The game never goes into enough detail to say for sure. Maybe verbal components do require calling out your spells by name. But that’s not the impression that I get.
But if you have assumed an alternate form via Wild Shape and now possess a beak and paws – your have no mouth or hands to become anything else any more. - Or maybe you wildshaped into a snake and have no limbs at all. In the spell it doesn't say that you grow claws … So I argued that you have to have a humanoid body to make this spell work.
The game has rules for spells that can only apply to humanoids; they say things like “Target: One humanoid creature,” or “This spell can only be cast on [or by, for personal spells] creatures with humanoid bodies,” or the like. Plenty of spells like that exist in the rules, like charm person, and non-humanoid races explicitly call out the inability to be targeted by such spells as an effect of having a non-humanoid creature type. So reading this requirement into the rules, despite it not being explicitly called out, is a misreading I think. The rules’ authors knew to write such requirements where they were intended. The lack of any such explicit requirement is telling in this case.
Beyond that, though, the rules we do have already cover much of your concerns. No, an effect granting a claw attack does you no good if you have no limbs with which to make it. It is—infuriatingly—not an explicit, written rule, but it is apparent in examples of both monsters and in effects like girallon arms that work their way around it. And you can’t use both a bite and a beak at the same time—the rules explicitly state that you have to use the “best” effect (which is presumably up to the person controlling the effect to decide which is best). But the rules also explicitly state that both effects are active; the spell certainly doesn’t fail wholesale.
And the rules generally try to have effects do as much as possible or reasonable, in cases where they are partially invalid. Magic effects aren’t supposed to mess with other magic effects unless they explicitly say so, for the most part. So saying you can’t target yourself with bite of the were-X at all because of wild shape is very inconsistent with that trend—to the point that I would say that it is wrong to rule that way. It certainly doesn’t improve the game.
I have the strong feeling, the spell wasn't meant to be combined with Wild Shape. It seems to be overpowered to turn into a Dire Bear and cast Bite of the Weretiger, giving you a strength score of 31+12=43 (which is that of a Titan...).
Certainly, I feel very strongly that if they meant the spells to be limited to humanoid casters, they would have said that. They did for plenty of other spells, and didn’t here. And the bite of the were-X spells were very much written for druids—it would be very odd if they forgot about wild shape while writing them.
But balance concerns are different from what the rules say, or what the rules are meant to say. The authors don’t intend things to be imbalanced, of course, but they may well have intended things to work a certain way, not realizing that they were overpowered. And they also may well write things not realizing that it deviates from their intent, potentially causing imbalance. Most of the discussion so far has been about what the rules actually say, and possibly what the authors meant to say, and not about what’s balanced.
So, is it imbalanced to wild shape into a dire bear and then cast bite of the weretiger and get 43 Strength? The druid needs to be 12th level to wild shape into dire bear form, and then needs a 5th-level spell to cast bite of the weretiger, when they probably only have 4 slots for that. More importantly, the druid needs a standard action, and then the effect only lasts 12 rounds—in a lot of situations, it won’t last long enough for the druid to cast it ahead of time, and won’t have the time to cast it mid-combat. And 5th-level spells are very, very powerful—there are definitely more powerful things that the druid might cast.
The counterargument is that it’s vastly stronger than anything a fighter or barbarian is doing. Which is true—but it’s true of nearly anything the druid does. The druid is one of the most powerful classes in the game, and the barbarian and fighter are some of the weakest. On some level, one concerned with balance should be glad that a druid is taking on the shape of a dire bear, casting bite of the weretiger—because there are absolutely much more powerful forms they could take, and vastly more powerful spells they might cast.
So my answer to the balance argument is kind of a shrug. It’s a druid. They’re overpowered. Wild shape is overpowered on its own and it’s a distant second to the druid’s spellcasting. If you’re concerned about it, consider nerfing the druid. Player’s Handbook II has a shapeshifting variant which replaces both animal companion and wild shape with a far weaker effect, that is also incompatible with Natural Spell. Doesn’t really stop the druid from being overpowered—again, it’s the spells that do that—but it does really limit the “add on” power of the druid class. You can still combine bite of the weretiger with shapeshifting (just cast bite of the weretiger first), but it’s harder.