However, the way that most articles (and even physicists) use the term "entanglement" is in a loose way. Within a good approximation, a multi-component quantum state can be represented as a tensor product of single component quantum states. When this isn't a good approximation anymore, you can say that the entanglement of the system is much more apparent.
To give an example of what I mean, there's a technique in quantum chemistry known as density functional theory (DFT), which is used to compute ground state energies of various molecules. For some molecules, it works pretty well. The benefit of DFT is that it is a fast calculation technique (well, as far as quantum chemistry goes), but it's speed comes at a price. Rather than using the Coulombic interaction of every pair of electrons to compute the energy of a molecule (or more technically, using every Coulombic interaction as terms in the Hamiltonian), a probability cloud of electron positions is computed instead, and this cloud is used as the energetic term.
This works really well for a lot of systems, but in some cases (superconductors, metals, solid-state physics, van der Waals interactions), this approximation falls apart because the effect of electron-electron correlation significantly affects the energy of the system. In this case, the full, non-separable wavefunction is required. The fact that this full wavefunction cannot easily be broken into an approximation of simpler, single electron wavefunctions means that the entanglement of the system is very apparent.