I think $2$ is not special, we just see the weirdness at $2$ earlier than the weirdness at odd primes.

For example, consider $\operatorname{Ext}_{E(x)}(\mathbb{F}_p , \mathbb{F}_p)$ where $E(x)$ denotes an exterior algebra over $\mathbb{F}_p.$ If $p=2$ this is a polynomial algebra on a class $x_1$ in degree $1$ and if $p$ is odd this is an exterior algebra on a class $x_1$ tensor a polynomial algebra on $x_2$. I say these are the same, generated by $x_1$ and $x_2$ in both cases and with a $p$-fold Massey product $\langle x_1,\dotsc,x_1 \rangle = x_2.$ The only difference is that a $2$-fold Massey product is simply a product.

In what sense are the $p$-adic integers $\mathbb{Z}_p$ the same? One way to say it is that if you study the algebraic $K$-theory of $\mathbb{Z}_p$ you find that the first torsion is in degree $2p-3$. If $p=2$ this is degree $1$, and $K_1(A)$ measures the units of $A$ (for a reasonable ring $A$). If $p$ is odd it measures something something more complicated. Another way to say it is that $\mathbb{Z}_p$ is the first Morava stabilizer algebra and there is something special about the $n^\text{th}$ Morava stabilizer algebra at $p$ if $p-1$ divides $n$. If you study something like topological modular forms, this means the primes $2$ and $3$ are special.

The dual Steenrod algebra is generated by $\xi_i$ at $p=2$ and by $\xi_i$ and $\tau_i$ at odd primes. But really it is generated by $\tau_i$ with a $p$-fold Massey product $\langle\tau_i,\dotsc,\tau_i\rangle = \xi_{i+1}$ at all primes, after renaming the generators at $p=2$. (Again a $2$-fold Massey product is just a product.)

I could go on, but maybe this is enough for now.