Lobsters show negligible outward senescence. Older individuals continue to moult, grow larger and produce more eggs than younger ones, and telomerase activity, which in many animals declines with age, remains high in lobster tissues. Those facts are the honest core of the immortality story.
The story fails at the moult. Each shed requires the animal to extract itself entirely from the old exoskeleton, including gill and gut linings, and then rebuild and mineralise a larger one. The cost scales steeply with size, and large old lobsters increasingly fail partway through, dying in the shell or from exhaustion and infection afterwards. Shell disease and the accumulated damage of repeated moults compound the risk.
Ageing a lobster is genuinely hard, because the exoskeleton, including any structure that might record annual growth, is discarded at each moult. Estimates of maximum age around a century exist but rest on growth models and indirect markers rather than counted years, so they should be read as approximations.