I have a theory on this that Andalite names are constructed similarly to how Roman naming works.
Basically the structure of a Roman name goes:
praenomen (personal/given name) nomen (family or clan name) cognomen (indicated a branch of a family, or was sometimes an honorific or given name)
(Roman names could also include multiple cognomens or get more complicated than this, or even change, but this is the basic structure so I won't go into more detail.)
So you have someone like Gaius (personal) Julius (family) Caesar (family branch).
As KingAlanI has pointed out, you can see a similar structure in the one family we have four names for. Elfangor obviously inherited middle (probably family name) Sirinial from his father Noorlin-Sirinial-Cooraf.
So why isn't Ax named Aximili-Sirinial-Isthill? We know that the Andalites sometimes limited couples to only having one child. It may be possible that in cases where there is a single female child of a family (in this case the Esgarrouth family), when she has children, a second child may be named for the mother's family rather than the father's. I would think this would come from earlier in the Andalites' history as more of an exception than a rule, which would have become more common as the population expanded and many couples would only have a female child. Or it may simply be that due to high mortality rates or something earlier in Andalite history, they adopted this system of naming after the father and then the mother. (The Andalites just don't strike me as being that pro-female to have such a naming system be common except to continue on male lines that might end.) The fact that the named siblings we know of in the books who are both (sorry Barafin) given full names corroborates that the second child is probably usually named for the mother, or other parent if her family name was used first.
If we had more families whose names were given besides sibling relationships, or if Seerow's full name had been given (or any of his family besides Aldrea) it would be much easier to tell if my theory is correct or make other guesses. Maybe if the first child is female, she is named for the mother's family. Would Seerow also have the middle name Iskillion, or something else entirely?
As for the mysterious last name, it is obvious since we don't seem to have any family members who share a last name, even between one parent and one child, that it is not used to signify a family branch in the same way that the Romans used.
However, it may be the family name of a female grandparent. So Elfangor and Aximili come from the line Sirinial through their father and paternal grandfather, the line Esgarrouth through their mother and maternal grandfather, the line Shamtul through their (possibly paternal?) grandmother, and the line Isthill through their (possibly maternal?) grandmother. I'm basing the priority of the father's mother's line over the mother's mother's line on their seemingly patrilineal naming, but that is not necessarily the case.
So that would explain the middle and last names sometimes being the same; this would mean that Harelin-Frodlin-Sirinial, for example, might be named for a grandmother who comes from the same family as Elfangor and Ax's father, by my theory. (Though how close the relation is could still be very distant. After all Sirinial and Esgarrouth might be Andalite equivalents to Smith or Williams.)
This naming pattern (especially with multiple children in the family) makes it easy to figure out related families. When multiple children were/are more common, the naming structure could be more random, or with multiple children sharing the same middle and last names.
tl;dr Why can't we have more named Andalite families to figure this stuff out??
I haven't reread the books recently so a lot of this is coming from just staring at Andalite names on wikis for a long time. x_x
