Could brothers or close kin join the same contuburnium?

[Nathan Ross]

there is that scene of father and son in opposing legions during a civil war situation in the first c. AD (the son killing the father).

Yep - Tacitus Histories 3.25. Although Tacitus adds a note to explain how this happened - the father Julius Mansuetus was from Spain, but had previously enlisted in XXI Rapax, based in Germania (which is interesting in relation to early principiate recruitment patterns!). His son, however, was enlisted by Galba in the new Spanish legion VII. Presumably the two had not seen each other for many years when they met on the battlefield.

The explanation at least implies that it would have been more normal for fathers and sons to end up in the same legion. Caesar's Gallic War (5.35) mentions that the primus pilus Quintus Lucanius was killed in battle trying to save his son, who was presumably serving in the same legion.

There are many inscriptions to soldiers described as brothers or even parents - these are not necessarily blood relations, but it seems common enough to suggest that it was quite usual for men of the same family to serve together.