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According to Mermin-Wagner theorem ferro/antiferromagnetic ordering cannot occur in ideal one dimensional systems. However there exists quasi-one-dimensional ferromagnetic materials attributing its long range magnetic order stability to magnetic anisotropy.
My question : How does magnetic anisotropy cause stable long range magnetic ordering in quasi-one-dimensional solids?
At least in the example you picked up to illustrate your question, the key to the answer might be here:
"The a axis, perpendicular to the chains in the structure, is the magnetic easy axis, while the chain axis direction, along b, is the hard axis."
This means that the effectively infinite magnetic moments corresponding to each infinite one-dimensional chain are in a good situation to achieve an (dipolar) Ising-type ferromagnetic coupling between each other, especially among neighbouring and nearest-neighbouring ones.
To complete the answer we need to consider how the situation would change for perfectly isotropic one-dimensional ferromagnetic chains. In that case, the total spin of each chain would be free to point in any direction. Without geometric restrictions, one can think that the magnetic energy would either be minimized by closing the magnetic field lines locally, or so weak that moments effectively are oriented randomly, with no long-range order, like in a paramagnet.
As complementary readings to understand the role of through-space interactions between magnetic dipoles, I recommend the case of dipolar Ising ferromagnet LiHoF$_4$. In this very anisotropic material, magnetic order can be achieved at low temperatures, even though the spins are zero-dimensional, in the sense that both direct exchange and superexchange between nearest neighbours are negligible. There are maybe hundreds of studies on this system; let me pick this one and we'll go back and edit the answer if this particular example seems useless:
The ferromagnetic transition and domain structure in LiHoF$_4$
I have not read the paper your linked to in detail, but I can offer some general answers to your questions.
...there exists quasi-one-dimensional ferromagnetic materials attributing its long range magnetic order stability to magnetic anisotropy...
(emphasis added)
We're talking about quasi-1D materials. Usually this means that it's a 3D crystal structure, but the strongest interactions along one direction. You can picture it like a bunch of spin chains arranged in a plane. At high temperatures, the interactions between the chains are negligible, but at low temperatures, those interactions can matter.
Anisotropy: The Heisenberg Model has full continuous 3D rotational symmetry, but the Ising model has discrete 2-fold ($Z_2$) symmetry. You can imagine a continuum between these two extremes: $$ H = -J\sum \limits_i \left[ g S^z_i S^z_{i+1} + (1-g)(S^x_i S^x_{i+1}+S^y_i S^y_{i+1}) \right] $$ For $g=1$, it's the Ising model, for $g=0$, it's an XY model, and for $g=1/2$ it's the Heisenberg model. An easy-axis ferromagnet ($1/2 < g <1$) prefers to order along the z-axis, so it acts a bit more like an Ising model. Therefore, you only have to break a discrete symmetry (glossing over some more subleties).
