There is no equivalent in AES to the Elliptic Curve *group* used in Elliptic Curve Cryptography. In particular there is no match for points with coordinates obeying a curve equation, or for a fancy rule to add these.

The parallel with ECC stops at AES using a finite field for bytes as does ECC for each point coordinate. In AES, the field is $\operatorname{GF(q)}$ with $q=2^8=256$. In ECC the field is $\operatorname{GF(q)}$ for some much larger $q$ (typically with hundreds rather than 9 bits).

One can think a finite field as a finite analog of the set of reals $\mathbb R$ (or of the fractions $\mathbb Q$) when it comes to algebra restricted to addition, multiplication, taking the opposite or the inverse, and testing equality (rather than order). A set with $q$ elements can be made a field if and only if $q=p^m$ for $p$ a prime and integer $m>0$. When $m=1$, the field $\operatorname{GF(p)}$ with prime $p$ is the familiar $\mathbb Z/p\mathbb Z$, also noted $\mathbb Z_p$, or equivalently integers in $[0,p)$ with field laws addition and multiplication modulo $p$. Such field is used in ECC for so-called prime curves like secp256k1 (with $p$ a 256-bit prime). But ECC works for any large finite field. E.g. sect283k1 uses field $\operatorname{GF(2^{283})}$, and this Elliptic Curve group uses field $\operatorname{GF}(9767^{19})$.

When $m>1$, including when $p=2$, a field element can be thought as a vector or tuple of $m$ elements of the field $\operatorname{GF(p)}$, or equivalently as the $m$ coefficients of a polynomial $P(x)$ of degree less than $m$ and coefficients in $\operatorname{GF(p)}$. Addition in the field $\operatorname{GF(p^m)}$ is addition of vector/tupple components in the field $\operatorname{GF(p)}$, or polynomial addition. When $p=2$ that reduces to XOR. See this for why the representation as coefficients of a polynomial makes sense to neatly define multiplication.

(In AES) $\operatorname{GF}(2^8)$ is an extension field of $\operatorname{GF}(2)$ (…) Does it just mean that each byte contains 8 bits (each bit being an element of $\operatorname{GF}(2)$) ?

It means that, *and* $\operatorname{GF}(2^8)$ is fitted with two internal laws (operations) that make it a field: addition that reduces to addition of each of the 8 components in $\operatorname{GF}(2^8)$, and a suitable multiplication.

Likewise what do the sub-fields $\operatorname{GF}(2^2)$ and $\operatorname{GF}(2^4)$ represent here?

They are different fields with 4 and 16 elements rather than 256. Sometime it might be interesting to represent an element of $\operatorname{GF}(2^8)$ as two elements of $\operatorname{GF}(2^4)$ or a four elements of $\operatorname{GF}(2^2)$. For addition such representation works quite directly, but multiplication is a more complicated story. That's not required in a standard implementation or study of AES (I've only seen it used in optimized implementation of the AES S-box).