Separately Created Weapons Don't Interact, and That's What These Are
Were these written as bonuses to unarmed strikes or to a PCs existing natural weapons then they would stack. Instead they seem to both be written as individual natural weapons, which would not stack. This is not as crazy as it sounds. Alter Self just says "you grow fangs" it doesn't say that they have anything to do with other fangs, teeth, etc. you may have. Your freaky Alter Self-ed form is free to have two sets of fangs. In fact, it seems, as a matter of straightforward English that fangs could not grow into fangs since they are already fangs, so the two sets of fangs are entirely separate.
Even if they are the same weapon there are precedents for making multiple types of attack with the same weapon, a Polearm Master's d4 bonus attack for example, so that still would not establish interaction.
They Do Otherwise Have Simultaneous Effects.
Remember that, like the Polearm Master's two types of attack, this has one (the Shifter one) as a Bonus Action attack. The Shifter effect only applies to this special bonus action attack. The other (the Alter Self one) applies to any attack, be it Attack Action, Attack of Opportunity, etc, you choose to make with that natural weapon. You would, however, once again be using the separate Longtooth weapon for the special bonus action attack, not the Alter Self weapon.
Furthermore Alter Self reads:
Your unarmed strikes deal 1d6 bludgeoning, piercing, or slashing damage, as appropriate to the natural weapon you chose, and you are proficient with your unarmed strikes.
This particular line mentions an effect on how the caster's unarmed strikes work without, by a literal reading, strictly limiting it to those made with the natural weapon created. Thus an Alter Self-ed character could make d6 unarmed attacks with hands or such even if it was the teeth that they changed into fangs, and this would apply in shifted form as well. This is not as illogical as it seems since there is nothing saying one is not more generally altered by the spell, but still probably not the intention of the spell. Thus were one to choose to make normal unarmed strikes for some reason despite having these swell natural weapon alternatives, one would still have proficiency and a d6 damage dice.
Note that it was errata-ed that everyone is proficient with unarmed strikes, which makes the mention of that in the Alter Self spell have no operative effect. Were this not the case it would give you proficiency you would not otherwise have on the Shifter fang attack.
There is An Arguable Reading Under Which These do Partially Stack, But it Doesn't Matter
The shifter passage reads:
If you hit with your fangs, you deal piercing damage equal to 1d6 + your Strength modifier, instead of the bludgeoning damage normal for an unarmed strike.
A very literal reading would say this applies to all attacks with fangs that belong to you, hence including Alter Self fangs. However this has no operative effect since your Alter Self fangs would already work this way, in addition to their +1-ness and Magic-ness.
To address a side note in the question, you can actually get Alter Self now if you give up an existing any school spell. Since it doesn't get you what I think you hoped for I'm guessing this is a lower priority, but Alter Self is a lot of fun outside of combat, and even in combat if you want to be a tooth-and-claw fighter it would be the way.
