I doubt that cable is for sub-sea use, almost all such cables have at least one layer of steel, and much thicker envelopes.
No, they don't. A major design constraint is the need to fit as much as possible cable on a cable-laying ship, and that precludes steel armour.
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The TAT-7 cable was steel core, […]
I didn't say "steel armour", I said "layer of steel", because you are quite right that at depth that is usually a steel "strength member" at the center of the cable.
The steel, or on a few very early cables: bronze, is necessary to prevent the length of cable hanging from the laying ship to the bottom of the sea from being stretched or pulled apart.
Air-dielectric coax-cables, like the one on the picture, has the additional weakness of a very fragile cross-section, which is why they were almost never used for sub-sea applications.
Summa summarum: The cable on the picture has no strength-member, steel or otherwise, and the very thin jacket offers insufficient mechanical protection against compression of the coaxial members, so that is clearly not a cable for sub-sea use.
In fact, I'm not even sure it is a cable for direct burial, it might be designed for use only in conduit.
PS: If you want to to read about the challenges of running coax under the Atlantic, the Bell Systems Technical Journal has excellent articles from the entire history of that practice.