Could a system of black holes form a "black" structure that has physical dimensions? For instance, could a series of black holes in the same orbit around an object form a black torus?
In 4D General Relativity there are uniqueness theorems that prove the No Hair conjecture for stationary, axially symmetric electrovac Black Holes.
That is, that in vacuum the only solution possible is the Kerr Newman solution, with only mass, angular momentum and charge as parameters (and the spherically symmetric Schwarzschild solution is a special case with no angular momentum nor charge). There are some constraints such as asymptotic flatness. A magnetic charge turns out is also allowed, though not physical. This was arrived at through a lot of work in proving the uniqueness while insuring that some strange conditions (such as multiply connected horizons) don't affect the proofs, mainly for the rotating case (Israel provide it for the spherically symmetric case early, it was harder with rotation)9999
But also see the review in https://nms.kcl.ac.uk/david.robinson/web_page/blackholes.pdf for other additional treatment for Black Holes in higher dimensions and with various matter fields.
For instance it is known that in 5D the Kerr Newman solutions are not unique as the only Black Holes possible. Also, it is known that for instance a Yang Mills field in General Relativity will also allow other, non Kerr Newman solutions. Work continues on finding different Black Hole solutions, including those from string theory, gauge theories, and with thermodynamics considerations due to the findings that the entropy is found in the horizons. I presume that Fletcher's hyper-toroid is one of those though I've not read the specific paper.
Needless to say most astrophysical work on Black Holes uses the Kerr Newman solution for the stationary or equilibrium Black Hole, and uses a perturbation formalism for the accretion of matter or other fields. That of course doesn't work in the strong gravity case such as for colliding or merging Black Holes, but of course the entropy theorems of Hawking for Black Hole thermodynamics allows you to treat the equilibrium states after the collisions or mergers. Numerical techniques were also used in analyzing the strong gravity phases of the Black Hole mergers seen by LIGO and announced in 2016.
The construction of gravitational wave interferometers will allow for better measurements on the next set of detections, with more accurate location of the events and also a wider frequency range. But we should be able to see inside neutron stars and tighter in and further out on mergers. It'll still have to wait for the space based interferometer planned for the 2020's (and funding dependent) for the large baselines needed to detect lower frequency, higher wavelength gravitational waves from the early universe, and large structures like galactic center and possible cosmological anomalies, and allow us to see behind the age of recombination 380,000 years after the Big Bang.
I have no expertise in this area, but it seems to me that a such collapse into a ring would be highly unstable, similar to Rayleigh-Taylor instability in fluid dynamics. The slightest perturbation would increase the deviation from symmetry exponentially, triggering the ring to break up into individual black holes, which may subsequently merge to form a small number of large black holes.
So while a black torus might be an ideal mathematical solution to the GR equations (or whatever equations are being used), I think it is likely to be virtually impossible in reality. Like balancing a pencil on its sharpened point : perfectly possible in theory, practically impossible in real life.