First, as with everything, pushing the limits of wording or definitions in the rules to gain some kind of advantage is subject to the approval of your DM. It's a fundamental part of D&D as a game that the rules are incomplete and in some places ambiguous or unspecified, and the DM's job includes navigating those ambiguities when they arise in order to create a fun experience for all of the players (including themselves). If your DM wants to let your Eldritch Knight attune to an ephemeral body part, great. But it's safe to expect that they won't.
Second, "weapon" is used in at least two senses in the 5e rules.
"weapon" in its usual English meaning of "a tool for fighting"; it's a category of equipment, with several sub-categories (martial weapons, ranged weapons, etc). One key feature of weapons in this sense is that characters can become proficient with them. As with many English words, it is also used in many semi-metaphorical senses, where it may via modification or context refer to things that are not literally weapons under this definition.
"weapon" as a category of attacks, to differentiate from "spell". A "melee weapon attack" may not involve weapons (a punch is such an attack), but it's distinct from a "melee spell attack".
So, what's a "natural weapon"? My reading is that a "natural weapon" is a weapon (in the first sense) insofar as it can be used to make weapon (in the second sense) attacks that do damage beyond what a regular unarmed attack would do. However, "natural weapon" is not a weapon in the sense that it constitutes a piece of equipment. The descriptions of the traits for player character races which provide natural weapons support this:
Your horns are natural melee weapons, which you can use to make unarmed strikes.
Your hooves are natural melee weapons, which you can use to make unarmed strikes.
your claws are natural weapons, which you can use to make unarmed strikes.
Your fanged maw is a natural weapon, which you can use to make unarmed strikes.
And, of course, the wording of alter self uses the same pattern:
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 strike.
In all cases, attacking with a natural weapon is an unarmed strike — the weapon is not a piece of equipment that the character wields. Alter self makes this even more explicit: the spell gives you proficiency with your unarmed strike, not with the natural weapons it creates. If natural weapons were fully "weapons" in the general sense, you would need to be proficient with them (not just your unarmed strike) to use your proficiency bonus on attacks.
Compare to the language of the Weapon Bond feature:
The weapon must be within your reach throughout the ritual, at the conclusion of which you touch the weapon and forge the bond.
While a natural weapon would meet these criteria, the rule reads as if it assumes that the weapon is a piece of equipment separate from you. My inference is that the intent of the Weapon Bond feature is to bond to a piece of equipment, and not a body part.