ExplanationThe two following texts
do answer the question, but in medically descriptive fashion.
The simplest explanation of "why" holds that natural selection has evolved an organ that is so critical to the body's functioning that, were it to fail, death would result.
Given the liver's criticality, and the fact that the body possesses a single liver, the liver also evolved the ability to regenerate. Mechanically complex organs (the heart) have not so far evolved such regenerative ability - perhaps natural selection can only go so far.
The clue to the liver's ability is homeostasis - "feedback" mechanisms that preserve function. Think of blood sugar - it has a desirable concentration which is regulated within narrow limits by insulin (to preserve the upper limit) and metabolisation of body fat and protein (to preserve the lower limit).
Now that we know "why", the answer for "none of the others can" - better asked as "none of the others were driven to" - is given by the existence of paired organs (lungs, kidneys, so we have a backstop), mechanical complexity (heart, too difficult) or "you-only-get-really-sick-and-die-but-it-often-happens-after-your-reproductive-peak" (pancreas, where
Diabetes can be survived for many years).
Remember that natural selection gets an organism to the point where it can reproduce - it "doesn't care" about old age - regrettably.
Medical descriptions.The liver is composed of a certain type of cells with increased ribosomal density, which mean the liver is the body's filter. We are all born with stem cells which are one of three types, Totipotent, pluripotent, or multipotent. The majority of stem remaining in humans after differtiaiton is multipotent cells, which have the tendency to become liver cells, so when lacking, the liver "recruits" some undiffertiated stem cells to become liver cells, and repair the liver.The liver can precisely regulate its growth and mass. Surgical resection of hepatic lobes or hepatocyte loss caused by viral or chemical injury triggers hepatocyte replication while enlarged liver mass is corrected by apoptosis. Hepatocytes have a great replicative capacity and are capable of repopulating the liver. However, "stem-like" cells proliferate when hepatocyte replication is blocked or delayed. Detailed studies of the mechanisms that regulate liver growth have been done in animals subjected to partial hepatectomy or chemical injury. Substantial progress has been achieved using appropriate transgenic and knockout mouse models for this work. Gene expression in the regenerating liver can be divided into several phases, starting with expression of a large number of immediate early genes. Hepatocytes need to be primed before they can fully respond to the growth factors HGF (Hepatocyte Growth Factor), TGFα (Transforming Growth Factor Alpha), and EGF (Epidermal Growth Factor) in vitro. Priming requires the cytokines TNF and IL-6 in addition to other agents that prevent cytotoxicity. Reactive Oxygen Species and glutathione content can determine whether the TNF effect on hepatocytes is proliferative or apoptotic. At least four transcription factors, NFκB, STAT3 (which are strongly induced by TNF), AP-1 and C/EBPβ play major roles in the initiation of liver regeneration.
it is apparent that liver regeneration involves the interplay of mitogens and comitogens ,paracrine and autcrine process and cellular communications and miscommunications.there is no single instead group of events causes liver cell regeneration