What you get is that if j:V to M is the ultrapower by any ultrafilter U on any set X, then every element of M has the form j(f)([id]). You can prove this by building an isomorphism from the ultrapower to the sets of this form. This way of thinking is also known as "seed theory".

To answer your specific question, if U is a non-normal ultrafilter on a measurable cardinal kappa, then it is not necessarily true that kappa generates all of M in that sense. In fact, this is true precisely of the ultrafilters that are isomorphic to a normal measure. For example, if U is normal, then UxU will not have the desired property.

What you get is that if j:V to M is the ultrapower by any ultrafilter U on any set X, then every element of M has the form j(f)([id]). You can prove this by building an isomorphism from the ultrapower to the sets of this form. This way of thinking is also known as "seed theory".

Theorem. Suppose that j:V to M is an elementary embedding of the universe V into M. Then j is the ultrapower map by a measure on a set if and only if there is some s in M such that every element of M has the form j(f)(s).

That is, the ultrapower embeddings are precisely the embeddings whose target is generated by a single element.

Proof. If j is the ultrapower by U on X, then let s=[id], and argue that [f]_U is j(f)(s). Conversely, if M = { j(f)(s) | f in V }, where we assume that f is a function on some set X such that a ∈ j(X), then define the measure U by A ∈ U iff s ∈ j(A). This is a κ-complete ultrafilter on P(X). One can show that Ult(V,U) is isomorphic to M, by mapping [f]_U to j(f)(s). QED

Theorem. If U is a κ complete ultrafilter on κ with ultrapower embedding j:V to M, then every element of M has the form j(f)(κ) if and only if U is isomorphic to a normal measure.

Proof. You know the backwards implication. For the forward implication, suppose that every element of M has form j(f)(κ). In particular, β = j(f)(κ), where β = [id]_U is the seed for U. But also, κ = j(g)(β), where g(α) is the smallest ξ for which f(ξ)=α. Let μ = { X subset κ | κ ∈ j(X) } be the induced normal measure. Note that X in μ iff j(g)(β) in j(X) iff β in j(g^-1X) iff g^-1X in U. So μ is Rudin-Kiesler below U. Also, U is Rudin-Kiesler below μ since X in U iff f^-1X in μ. So μ and U are isomorphic.QED

One may illustrate the situation with product measures. Suppose that U is normal. The product measure UxU is isomorphic to the two-step iteration, where j_0:V to M is the ultrapower by U, and h:M to N is the ultrapower in M by j_0(U). Every element of M has the form j_0(f)(κ), and every element of N has the form h(g)(κ₁), where κ₁ = j_0(κ). If j is the composition of j_0 and h, then j:V to N and every element of N has the form j(f)(κ, κ₁). If one only looks at j(f)(κ) inside N, then you will only get ran(h), which is isomorphic to M, but not all of N. So this would be a counterexample to what you asked about.