... Can a family tree be viewed as an abstract form of a worldline? Why or why not?
The simple answer is no, for two reasons:
A tree is a more complex structure than a line (graph theory), and
The concept of a worldline was created primarily to address simple objects moving through mostly empty space. It is not broad enough to deal meaningfully with objects that interact in complicated ways.
Hermann Minkowski is the fellow who came up with worldlines:
... to avoid saying "matter" or "electricity" I will use ... "substance." We fix out attention on the substantial point which is the world-point x, y, z, t, and imagine we are able to recognize this substantial point at any other time... we obtain, as an image ... the everlasting career of the substantial point ... a world-line ... [over] t from $-\infty$ to $+\infty$. The whole universe is then seen to resolve itself into similar world-lines...
-- H. Minkowski, Space and Time (Cologne Lecture), 21 Sept 1908.
Since Minkowski's lecture was on Einstein's special theory of relativity, his goal was to provide a mathematical framework for what happens to objects as they approach very high velocities. His worldline definition thus focused on well-defined objects (small or large) moving mostly through empty space. Objects that change form or interact in complex ways, what we would now call condensed matter physics, were just not part of his agenda.
Put that all together and you get roughly this:
A Minkowski worldline is a graphical image of how a cohesive cluster of material particles (or just one particle) moves through mostly empty space over a long period of time.
But your question remains a fascinating one, because if taken seriously it forces a more careful examination of Minkowski's original concept.
For example, look carefully at this part of Minkowski's definition:
"... imagine we are able to recognize this substantial point at any other time ..."
What Minkowski was saying in this seemingly simple line is that to define a worldline you must first have a universe in which clusters of particles have uniqueness or identity over time. This "uniqueness" then allows a future observer (and did you just notice that it's not just quantum mechanics that requires observers?) to look and say "even though this entity is in a new location, it has a unique signature or identity that enables me to recognize it as the same one I saw earlier."
That's no small concept, and it is not a given in just any universe. A universe of nothing but diffuse gases lacks it beyond the atomic scale, for example. So does the world of quantum mechanics, where for example boson particles with identical identifying features (think photons in lasers) become literally and profoundly indistinguishable from each other.
In our much more interesting universe, however, objects such as tiny dust particles or grains of sand can persist for long periods. Their location alone provides a certain degree of identity, provide you remember to look at them often enough. But even more specifically, such objects exhibit unique and persistent patterns of particle types and relative geometries that make them classically identifiable over time.
Another way of saying that is that the ability of clusters to hold unique particles in unique configurations makes them into memory devices, that is, entities that are capable of holding and storing for long periods a more abstract quantity called "information" that in this case specifies the identity of the entity.
It is this more abstract information quantity that enables Minkowski's ability to "recognize this substantial point at any other time," and for that reason it merits a specific name for this discussion. So for convenience I'll refer to it as an "identity worldline" that moves in exact synchronization and location along with the material worldline of particles and mass that it labels.
Separating Minkowski and Identity Worldlines
Since Minkowski worldlines always conserve mass, they really are just lines. They cannot split or form more complicated graphical structures without violating their own definitions.
However, the identity worldlines that travel with the Minkowski mass worldlines are a much more interesting case. Since they are composed of information rather than matter, they can be replicated fairly easily to create tree-like branches. Since identity has no inherent mass, it can also move quickly and flexibly across space in ways that no material object ever could.
So even though the original massive particle worldline and its co-traveling identity worldline remain bound together until the particle is destroyed, copies of selected subsets of its identity worldline can split off at any time to form new branches on an identity worldtree.
In a superb bit of pre-computer irony, Minkowski inadvertently invokes the use of branched identity worldtrees when he asks the reader to "imagine we are able to recognize this substantial point at any other time." An entity can be recognized later only if it was both observed earlier, and if the memory of that observation was preserved independently of the object. This act of identifying and externally preserving identifying data about Minkowski's "substantial point" is identical to creating a new branch on its identity worldtree.
Families as DNA Identity Worldtrees
Now, at last, this long addendum finally get back to your original question: Can a family tree be viewed as an abstract form of a worldline?
In terms of Minkowski mass-conserving worldlines, the answer remains no.
However, in terms of the identity worldlines and worldtrees that travel with Minkowski worldlines and make them observable, the answer is that there are very close parallels indeed.
Long ago, life learned an amazing trick of splitting identity worldlines in a completely symmetric way. So instead of one branch holding an incomplete copy of the original identity, both branches contain essentially identical identity data, stored in essentially the same physical form. The object has been cloned with a degree of precision that rarely if ever happens with non-life.
The most common form of this life trick is a molecule called DNA, and it is of course this DNA that defines family trees.
And while it may seem strange for a discussion of Minkowski's concept of a worldline to end with DNA, it is in fact impossible to ignore the profound role that the only slightly broader concept of identity worldtrees plays in every aspect of physics. You cannot observe the physical world without creating branches on the identity worldtrees of the objects observed. At an even deeper level, every creation of such an identity branch moves the observed object a little more into the classical world of abundant historical information, and a little farther away from the curiously undefined and ahistorical world of quantum mechanics.