Aging and death of perennials
The issue is based on your perception of what an individual ought to be. The concept of individual is actually often hard to apply in biology and therefore is the concept of ageing. In other words, the wish to categorize is a rather human tendency but it may fail to make objective representation of biodiversity. The book The Major Transitions in Evolution Revisited makes a very good job to explain the difficulties to deal with the concept of individual in biology. 

You seem to be happy to consider an offspring be a different individual if you can still perceive the parent individual. In bacteria, both resulting cells could be considered as offspring and therefore new individuals. Note however, that asymmetrical division bring some more blur into these categorizations.

Note also that single cell bacteria do age. See the wikipedia article on bacterial ageing. You might also want to have a look at the wikipedia article on biological immortality as well as the posts Why is the Hydra Biologically Immortal? and Do crocodiles age?.