This issue is resolved the same way almost all disputes over semantics are resolved in physics: there are multiple definitions of the same words at play here, each of which are perfectly legitimate, and the seemingly contradictory statements are just based on different definitions.
Whenever you set up a calculation in physics, whether in Newtonian mechanics or in quantum field theory, you pick a subset of the universe to count as your "system". Everything else is treated as external, a "background" that influences the system but whose detailed state is not kept track of.
For example, consider a ball dropped near the surface of the Earth. The Earth and ball together have three-dimensional translational symmetry, so $\mathbf{p}_{\text{ball}} + \mathbf{P}_{\text{Earth}}$ is conserved. But in practice, you might not want to consider the motion of the Earth. Instead, you can treat the ball as your system, and account for its interaction with the Earth by adding a potential term $U(\mathbf{x}) = m g z$. But this term is not invariant under translations in the $z$-direction (i.e. from this perspective, the Earth has spontaneously broken this symmetry), so $p_{\text{ball}, z}$ is not conserved, though $p_{\text{ball}, x}$ and $p_{\text{ball}, y}$ still are.
So is momentum "really" conserved in this situation? Is there "really" translational symmetry? It's not a sharp question: there are just two separate translational symmetries one might want to consider, which correspond to different momenta. One is conserved, another isn't.
Can someone explain in what sense the momentum (and angular momentum) conservation is violated in crystalline solids but not in liquids?
When you have a sample of solid or liquid sitting in your lab, there's always a translational symmetry that corresponds to moving the sample around in your lab (neglecting the effects of gravity). This is a perfectly legitimate and important symmetry, because it tells us that we can do the experiment anywhere we want in the lab, and it tells us that the ordinary momentum of the sample is conserved.
But once you've fixed where the sample goes, and want to analyze the dynamics within the sample, this symmetry isn't useful anymore. Instead, when condensed matter physicists talk about translation, they mean a symmetry that translates the excitations of the sample within it, without translating the whole sample itself. For instance, in a solid you might translate the electrons without moving the atomic lattice, or in a liquid you might translate a sound wave within the liquid without moving the bulk liquid itself. The corresponding momentum-like quantity is called crystal momentum for solids (or more generally, quasimomentum), and for solids it isn't conserved because the interaction with the lattice isn't translationally invariant.
But isn't regular momentum still conserved? Absolutely. If you want, you can artificially separate, say, the ordinary momentum of a phonon from the regular momentum of the rest of the crystal lattice. When the phonon's ordinary momentum changes, the rest of the crystal's ordinary momentum changes in the opposite way -- it serves as a "reservoir" for ordinary momentum, just like the Earth serves as a "reservoir" that allowed $p_{\text{ball}, z}$ to change.
The situation isn't any different in particle physics. For example, the universe as a whole is still $U(1)_Y$ symmetric, and accordingly the hypercharge of the entire universe is conserved. But this fact is not particularly useful in constraining reactions that we can see. The reason is that $U(1)_Y$ is spontaneously broken by the Higgs field, and hence serves as a background reservoir of hypercharge, allowing the total hypercharge of excitations to change. We're so used to living inside this situation that we often summarize it as "$U(1)_Y$ is broken". Similarly, condensed matter physicists are so used to living, e.g. in a crystal lattice that they might just say "translational symmetry is broken".