This is not really answering your question. But it's worth pointing out that periodicity of Clifford algebras (closely tied to Bott periodicity) already gives you the "periodicity" in the explicit lower bound. If you want a sphere with $8a-1$ independent vector fields, you can take an irreducible representation $M(8a)$ of the real Clifford algebra $C(8a)$; the Clifford module structure provides that many vector fields on the unit sphere in $M$. The algebras $C(8a)$ are all Morita equivalent, and since $\dim_{\mathbb{R}}C(8(a+1)) = 16^2\, \dim_{\mathbb{R}}C(8a)$, you must have $\dim M(8(a+1))=16\, \dim M(8a)$, so we learn that $\rho(16^a-1)\geq 8a-1$.
As for the upper bound, this reduces to something about the real K-theory of truncated projective spaces, which Adams calculated, and which clearly has something to do with Bott periodicity.
But I would agree that this all sounds like a big coincidence. I wish I knew something better to say about it.
Here is one conceptual description of the relationship. (I wouldn't call it an explanation; I don't really know why it's true, except that it's because Bott periodicity is true.)
KO-theory is the first Weiss-derivative of the $K$-theory of Clifford algebras.
More precisely: given a real inner product space $V$, we get a category $M(V)=\mathrm{Mod}(Cl(V))$ of finite dimentional, $\mathbb{Z}/2$-graded modules over the real Clifford algebra $Cl(V)$. We can consider the $K$-theory space $K(M(V))$ of the topological category $M(V)$; take this to mean something like group completion with respect to direct sum.
Thus,
$$ M_n = \pi_0 K(M(\mathbb{R}^n)),$$
using the notation in your question.
We thus have a functor
$$
V\mapsto K(M(V))\colon \mathcal{L} \to \mathrm{Top}_*
$$
from the category $\mathcal{L}$ of finite dimentional inner product spaces and isometries, to pointed spaces.
Michael Weiss came up with an "orthogonal calculus", which produces a tower of functors approximating a functor $F\colon \mathcal{L}\to \mathrm{Top}_*$, whose $n$th layer takes the form
$$V\mapsto \Omega^\infty( ((D_nF) \wedge S^{V\otimes \mathbb{R}^n})_{hO(n)}),$$
where $D_nF$ is some spectrum equipped with an action by the orthogonal group $O(n)$.
If we take $F(V)= K(M(V))$, then it turns out that
- $D_nF\approx *$ for $n>1$, and
- $D_1F\approx KO$.
So $KO$ is the first Weiss derivative of $F$.
Why is this true? Weiss gives an easy formula for $D_1F$. Define
$$F^1(V) = \mathrm{ho.fib.}\bigl( F(V) \to F(V\oplus \mathbb{R})).$$
This turns out to come with a tautological map
$$a_V\colon F^1(V) \to \Omega (F^1(V\oplus \mathbb{R})).$$
It turns out that $D_1F$ is exactly the spectrum associated to the prespectrum $\{F^1(V), a_V\}$.
In our case, the spaces $F^1(V)$ are the spaces of the $KO$-spectrum (as in Karoubi, I guess), and the maps $a_V$ turn out to be the Bott maps. Which are equivalences by Bott's theorem.
The higher derivatives $D_nF$ are computed using the fibers of $a_V$, but as these are already contractible, $D_2(V),D_3(V),\dots$ must vanish.
Replace $Cl(V)$ with $Cl(V)\otimes \mathbb{C}$ to get $KU$.
This seems like a neat fact, though I've never been able to figure out what it's good for.
Best Answer
This problem was investigated by Cecile DeWitt-Morette et al. This is a review article describing the role of Pin groups in Physics. This article includes also a historical survey and a comprehensive list of references about Pin groups.
Becides the fact that the Clifford algebras Cl(3,1) and Cl(1,3) are nonisomorphic, there is the question if the two types can be experimentally distinguished. The article gives a positive in the neutrinoless double beta decay experiment where it was observed that the neutrino has to be a Pin(1,3) particle.
The article also describes solutions of the Dirac equation in topologically nontrivial spaces where the vacuum expectations of the Fermi currents are different in the two types of groups. (Thus one might conclude that this difference can be used to get information about the space-time topology).
The article also refers to Choquet-Bruhat, DeWitt-Morette and Dillard-Bleick's book about obstructions to the construction of Pin bundles.