When I first read about vacuum entanglement, I understood it in exactly the same way as Ron wrote in this post. As can be very clearly seen, in a free scalar theory and in Schroedinger wavefunctional picture, vacuum state is a not a product state in terms of position-space wavefunction, but is a product state in momentum-space, and even more trivial(in terms of product structure) a state in the Fock-space representation. They should be all mathematically equivalent, I suppose.

However, the question of whether "entanglement" is properly so called depends on how we identify subsystems that are in a sense "physical". Changing from position-space wavefunctional representation to the momentum-space one, or to the Fock-space one, is clearly not a change of basis of the Hilbert space, but a change of identification of subsystems.

To elaborate my point, consider a scenario in ordinary quantum mechanics, two spin-$\frac{1}{2}$ particles positioned at two points, are in a entangled state $|+-\rangle-|-+\rangle$. However, we can relabel the basis of the 4-dimensional Hilbert space as

\[|A\rangle:= |+-\rangle-|-+\rangle\\|B\rangle:=|+-\rangle+|-+\rangle\\|C\rangle:=|++\rangle\\|D\rangle:=|--\rangle .\]

In this case, it is surely absurd to say our state is not entangled because it can be written as a single ket $|A\rangle$, because it is clear what must be identified as the physical subsystems.

The scalar field theory I discussed seems to be an infinite-dimensional analog mathematically, but in the field theory case I'm no longer sure what should be identified as the physical subsystem.

In fact, in the post I linked, twistor59 raised the same question under Xiao-Gang Wen's answer, I quote,

as you say, the vacuum isn't a product state, but I was curious about what the subsystems were in order even to discuss whether it is a product or not.

but I see no satisfactory discussion in that post.