Do all animal cells have lysosomes?
You are probably referring to the process of phagocytosis, where macrophages engulf a pathogen by trapping it inside a vesicle called "phagosome", which then fuses with a lysosome, resulting in the digestion and disposal of the pathogen.

But in fact all eukaryotic cells have lysosomes. This is of course biology we are talking about, so there are many subclasses, different names, specializations and exceptions to the rule.

In plant and fungal cells, we find vacuoles, which serve a similar function as lysosomes in animal cells. There are also "lysosome-related organelles" with specialized functions in specific cell types (Marks et al. 2014).

The main role of lysosomes has been described in the recycling of macromolecular material. Their lumen has a low pH level (below 5) and contains a lot of enzymes specialized in degradation of e.g. proteins, nucleic acids etc. They can digest cellular waste products and material taken up by endocytosis. Nutrients can be reabsorbed into the cytoplasm.
Furthermore, research has proposed additional roles in nutrient sensing and metabolic homeostasis (Perera and Zoncu 2016).

That lysosomes play a crucial role in many cell types can also be seen in lysosomal diseases, which often result in severe effects in multiple organs (Ballabio and Gieselmann 2009, Saftig and Klumpermann 2009). In these diseases, lysosomes fail to degrade their targets, which impairs their ability to clean out cellular waste.

Here is a summary from the book "The Cell".