We are also patriarchal and take on the names of our forefathers, however, when we descend from one man that had multiple wives... some differentiation is required -- hence the need to address the descendants of each family by the name of the matriarchal ancestor.
The 'last' names remain patriarchal; while being addressed by the name of your matriarchal ancestor is just convenient due to the branches that result in polygamous marriages
The guy you responded to made an error. We don't take on the mother's clan, it merely serves as a name for intra-sub-clan merger. None of those guys take on the mother's clan, but mention her clan as an addition to the clan they belong to for the realities of the standing of the clan. Basically they do it to establish precedence as a supplementary line on their father's clan to distinguish themselves and form a respective uterine estate that tries to distinguish itself from their ancestral uncle descendants that have different mothers but different wealth, territory, conflicting interests, etc.
This is an example:
"An example from the ‘Ali Geri primary lineage ofthe Dulbahante clan may serve to illustrate this. This lineage is segmented as shown and had (in 1956) a male population of approximately 2,800. It will be seen that the lineage is divided in the generation of ‘Ali Geri’s sons into three uterine groups, the Bah Helaawe, Bah Helaan, and Bah Ogaadeen. The Bah Helaawe and Bah Helaan are the successive issue of‘Ali Geri’s marriage to two sisters, both daughters of Lagmadoobe of the Bartirre clan. After the death of the first, ‘Ali Geri took her sister Helaan in soro' ratic marriage. These two uterine groups are joined together as the Bah Lagmadoobe and form one dia'paying group. The remaining three lin' eages, the Hirsi, Subaan,35 and Shoowe ‘Ali Geri, each of which is an independent dia'paying group, derive from ‘Ali Geri’s three sons by a woman of the Ogaadeen clan and are grouped together in opposition to the Bah Lagmadoobe as the Bah Ogaadeen, which, though divided (in 1956) into three separate dia'paying groups, a few years earlier had been a single dia'paying group."
They never adopt the clan of their mothers at all. Merely mentioning the clan she belongs to signifies more shared uterine relations by the clan progenitor and mother in polygamous relations, where other mothers give rise to children that form distinct agnatic descent groups.
This is actually not a matriarchy, the least. During the mother's life, she lived as a wife, she did not establish a clan at all. And in the "bah" grouping, only her patrilineal clan is mentioned, and neither is her clan taken on, as her male descendants only claim patrilineal descent and clan relations with the patriarch, i.e., the husband.
The Bah Helan who descend from Guuled, Warfaa and Buuraaleh (three sons of Ali Geri), don't say we descend from the Helaan clan. They say
Bah Helaan is Ali Geri. "Bah" here is an asterisk that is important. Be Helaan clan doesn't claim the actual Helaan clan, nor does the Helaan clan claim Bah Helaan. And you see, more Bah were created later, long after those mothers were gone, meaning it is a political arrangement.
This is a contractual alliance based on lineage coalition that is strictly patrilineal, a very common thing in clans and tribes in patriarchal systems, especially among groups that belong to pastoralist ideology, where territory, wealth, and power are significant for survival.
So it is indeed similar to how you described the Dinka do it.