There are several basic defining properties that distinguish elementary fermions - quarks and leptons - from other objects such as dogs:
- they are quantum mechanical entities that are able to interfere in double-slit experiments; they are indistinguishable
- these particles satisfy the rules of special relativity and the relativistic dispersion relations
- they carry spin and are created by quantum fields with a spinor index
- quarks carry color and they transform as a representation of a gauge group
- consequently, they interact through interactions mediated by Yang-Mills gauge fields
Elementary particles can't be dogs or octopi because dogs and octopi violate every single property listed above. If someone violates all these requirements, it means that he has made no (0) progress in describing these elementary particles. Let us see why the octopi violate these requirements one by one.
Uniqueness
Dogs and octopi can't realistically generate interference patterns in double-slit experiments. It is because they carry a huge entropy: a dog can be represented by many different microstates and these microstates don't interfere with each other. It is absolutely necessary for the quantum state describing an electron to be unique - determined by the position (or momentum) and spin only - in order for the electron to be able to interfere.
Elementary particles of the same kind must be indistinguishable and the fermions such as electrons must moreover obey the Pauli principle; otherwise chemistry could not work. Mammals violate these rules because their complicated internal structure makes them distinguishable.
A necessary condition for this uniqueness that is required for the existence of interference is the uniqueness of the vacuum itself.
Unusual theories that imagine a chaotic discrete structure at every point of "empty" space violate this rule maximally. A double-slit experiment would never create an interference pattern if vacuum had a large degeneracy, as envisioned by theories of "discrete gravity" and similar hopeless approaches to physics. The vacuum must be a completely pure, unique state, and the one-electron states must also be pure, well-defined states in the Hilbert space that are described by the position/momentum and the spin only.
No chaos is acceptable for particle physics. Some people, especially cranks, imagine that the deepest idea that they can dedicate to the world of physics is to make things complex and to give the particles complicated internal structure. But the experiments offer clean outcomes and require just the opposite: the vacuum and the particles must be very clean, unique, and if there is any internal structure, there must exist reasons why the structure is always found in the same state.
The difficult task for those who probe physics beyond quantum field theory is not how to make things complex, foggy, and chaotic: it is on the contrary how to make things consistent, pure, well-defined, and consistent with the very sharp and nearly accurate laws of Nature that we have already understood, experimentally verified, and summarized in the Standard Model and General Relativity.
Relativity
A related point is that the elementary particles can't be identified with distortions of some "underlying structure" that is not unique because such a picture would violate the fact that the elementary particles follow the rules of special relativity. Elementary particles can't be excitations of an aether - a discredited notion from the 19th century physics that some people are trying to revive under the name "spin network" - because such an aether would break the Lorentz invariance. We know from experiments that the Lorentz invariance is either exact or an extremely accurate approximation.
All people who say that the Lorentz invariance is something that can be ignored or something that is cheap to obtain are crackpots because special relativity is one of five most important and five most constraining discoveries of the 20th century physics.
Octopi that swim in the ocean do not respect the rules of special relativity (unless you include the details of the ocean into your description): water spontaneously breaks this symmetry. Everything else that resembles a complex object on a generic complex background is all but guaranteed to do the same thing. Particle physics follows very different rules from the ocean.
Spinors
Elementary fermions are excitations of spin-1/2 fields in spacetime - quantum fields that transform as a spinor under the Lorentz group. Again, this is no detail. It is a very defining feature of the leptons and quarks. Octopi and dogs don't transform as a spinor. A naive classical picture of octopi can't be compatible with the spinorial gauge theory of an electron.
Note that the correct spin of particles can be extracted from string theory in all of its realizations. For example, particles can be viewed as excited strings and the excitations themselves transform as the appropriate representation of the Lorentz group. It is because the excitations of the "minimal energy string" are operators themselves and they naturally transform as representations of various groups. A random octopus embedded in a complex environment won't transform as a representation of the Lorentz group - this group will be broken.
Color
Quarks also transform as the three-dimensional representation of a group, namely the colorful SU(3) group. The word "group" really means a "symmetry". A symmetry is a set of transformations that transform one object onto another object. It is something that shows that these two or more objects related by symmetries actually have identical physical properties.
If you draw three octopi that differ in details, they can't have the same physical properties, and consequently they cannot form a representation of a group. If three "colors" of quarks don't have physically indistinguishable properties, you won't ever be able to find an SU(3) theory that creates consistent forces between them. They will never have the right interaction terms with the gauge fields, even if you believe that such a gauge field could also be found.
Strings are the most complicated objects that can behave as elementary particles with the right properties. Branes of other dimensions are in principle capable to do the same thing. But it is hard to quantize branes of wrong dimensions directly - we know how to define quantum theories describing internal dynamics of branes using open strings stretched between them: a brane on which an open string ends is called a D-brane.
Generic animals can't play the role of elementary particles because of all the reasons above. Any paradigm that is meant to be treated seriously by theoretical physicists must explain why it reproduces the same kind of Lagrangians - or equations of motion - that we know from the Standard Model and/or General Relativity: quantum fields with Lorentz or spinor indices, color indices, and their right products in the Lagrangian. This task is very nontrivial which is why string theory is the only known way (and probably the only mathematically possible way) how to describe observed physics by something else than point-like particles or by quantum fields defined at each point of a spacetime.
