Yes & NO
As in tissue culture lines, the cell line that all subsequent propagules from the original cultivar will indeed be the same age. So I respectfully disagree with those that give a simple flat no. Yes, the genetic cell line, all being from a single source, are indeed the same age. All Pinus banksiana 'Chippewa' are a cell line that is the same age.
I raise orchids, a number of my orchids are individual clones from a single seed, some from seed that germinated over 100 years ago. Like an iris, or lawn grass, the orchid grows, divides into multiple meristematic growing points, spreads, the older tissue dies but the front leads are still genetically identical to the original. If you plant a golf green (where it is mowed often enough that once plants it is mowed too low to ever set seed) it is genetically identical 50 years later, even though no part of the grass plants are older than 2 or 3 years. New grows, old fades away. So like 'Pando' the giant 10,000 year old or older Quaking Aspen, yes, the tissue is indeed that old, even though none of the wood is older than a couple hundred years.
By the same token, sprout a maple or elm, root cuttings or make grafts. The tissue line obviously is as old as the original seedling, but the plant in front of you, if rooted from a cutting, or a 1st or 2nd year graft, will be functionally a new plant. So BVF is correct, a cutting from a Juniperus 'Kiushu' is for all intents and purposes a young cutting. Even though the tissue lineage is near 100 years old.
So for practical purposes - BVF is correct, a 2 year old cutting of 'Kiushu' is a 2 year old plant. Horticulture wise, this is a blanket truth.
Note - the exception is with regard to flowering. Malus, are well documented to need anywhere from 5 years to 25 years for the genetic clock to mature to the point of first flowering. Cuttings and grafted plants of old enough to have begun flowering, apples and crab apples (Malus sp), will flower immediately as soon as energy allows. So in that respect, age of the tissue culture lineage does show.
Now to your second question. DO TREES DIE OF OLD AGE - the answer is an emphatic NO. At least not in the same sense as mammals. The tables one sees in Forestry Management or Landscape Management books are based on utility, not on OLD AGE in the sense of vertebrate mammals. For forestry, the age of a stand of tree is economic, Senility or Old Age for forestry is when 50% of a stand (usually defined in acres or hectares) when 50% of the stand has trunks that show some heartwood rot. That is defined as the "death" of the timber acreage. It is an economic measure. You maximize board feet of harvest by harvesting before you have 50 % of your trees with heart rot. Yet we all know of oak trees, in which an individual tree will have survived 500 years beyond the point at which heart rot began. I've got a red oak in my driveway that has been hollow for at least 200 years. Its still growing nicely. Every now and then it drops a branch, I never park a car under it, yet it is beautiful.
In Landscaping Manuals - old age is when a tree has aged past its ornamental purposes. Mimosa is listed as short lived, maybe only 30 or 40 years, yet they can, and do live for more than 100 years. But after the first 20 years, they tend to suffer various rots, broken branches, and become profuse producers of root suckers, such that they become very messy, ugly trees. They will need to be removed because they simply are too ugly for a planned ornamental landscape. That tree is described as ''dead'' in that it needs to go. Similar apple trees in orchards and back yards are often removed on a schedule to maintain productivity. THe tree would happily persist a century longer, but due to pruning techniques, or older wood simply does not produce as much fruit as younger trees, so they get removed. Cherry trees are harvested mechanically, by vibrating the trunk. This vibration causes scarring in the cambium of the trunk. By the time the cherry tree is 25 years old, the trunk is so scarred from mechanical harvesting the cambium is no longer intact enough to support a good harvest. The tree gets replaced with a younger tree.
Trees have no fixed life span. THe aspen that has spread clonally is over 10,000 years. There are creosote bushes that are estimated to be clonaly over 1200 years old. IF you protect a tree from disease, and accidents of environment (lightning and fires) there is no finite limit as to how long a tree may live.
Does this help?