There are a number of images on the Web, most beautifully Visible Body's artwork, which show the caudate nucleus and putamen linked by regularly spaced connections across the barrier of the internal capsule. Since vertebrate forebrains are treated as having only a few segments (see Michael Carstens, 2017), which seem tough to relate to the C-shape of deep cerebral structures, there is something about such figures that piques curiosity. In a paper like Heide Hörtnagl et al., 2020 the situation seems less clear to me, but they weren't really focusing on this issue. Is there a regular/regulated spacing between the connecting bundles, and does it correlate with any other observable anatomical regularity?
These seem to be referred to in literature as "transcapsular" or "caudolenticular" (gray) bridges. My initial thought was that the picture you shared is just some weird artistic rendering but upon seeing histological slides these are quite prominent and familiar to me as part of the basic "look" of these structures in histology. You can see them in any coronal slice through this area, this atlas, "Cross-Sectional Atlas of the Human Head" by JS Park has them labeled though often they are not labeled.
(left image from JS Park; annotations on the right image are mine. There are many more "bridges" than I've drawn but I highlighted the most prominent to compare on the left side. Probably those deeper in the section relative to the camera are showing up a bit faint or blurred. This is also just one histological section, the bridges are found throughout the length of the area where caudate and putamen are located, at least a couple cm)
There may be other terminology, as well, but I'm having trouble finding it.
I see no reason or foundation on which to connect these to any sort of segmentation of the developing vertebrate forebrain. Since resources are sparse it seems difficult to find a reference, but my thought is that this is just an emergent pattern you get as different fiber pathways cross one another, like how your fingers look when you fold your hands together.
Some other references:
Mendoza, J. E., & Foundas, A. L. (2008). The Basal Ganglia. In Clinical Neuroanatomy: A Neurobehavioral Approach (pp. 153-193). Springer, New York, NY.
Nieuwenhuys, R., Voogd, J., & Van Huijzen, C. (2008). Brain slices. The Human Central Nervous System, 137-173.