I think 2 is not special, we just see the weirdness at 2 earlier than the weirdness at odd primes.
For example, consider ExtE(x)(Fp , Fp) where E(x) denotes an exterior algebra over Fp. If p=2 this is a polynomial algebra on a class x1 in degree 1 and if p is odd this is an exterior algebra on a class x1 tensor a polynomial algebra on x2. I say these are the same, generated by x1 and x2 in both cases and with a p-fold Massey product < x1,...,x1>=x2. The only difference is that a 2-fold Massey product is simply a product.
In what sense are the p-adic integers Zp the same? One way to say it is that if you study the algebraic K-theory of Zp you find that the first torsion is in degree 2p-3. If p=2 this is degree 1, and K1(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 Zp is the first Morava stabilizer algebra and there is something special about the n'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 \xii at p=2 and by \xii and \taui at odd primes. But really it is generated by \taui with a p-fold Massey product < \taui,...,\taui>=\xii+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.