Could one keep a shoggoth as a pet? The authorities in their blindness make no reference even to their existence, so there is certainly no legal barrier. But is there a practical one? The answer depends on which sort of shoggoth one speaks of.
Whilst the most well-known shoggoths are those described in Lovecraft’s At the Mountains of Madness (sometimes known as shaggoths), Bloch’s “Notebook Found In a Deserted House” provides evidence the name is also used by the Dark Young of Shub-Niggurath when speaking of themselves. In “A New Family Tree: Dark Young and Human Dreams” (Medieval Metaphysics Journal 74, 1981, p.55–89) LN Isinwyll provides evidence of Dark Young leading rituals since prehistoric time, accepting sacrifices on behalf of Shub-Niggurath, and providing revelation to worshippers; thus they are not merely intelligent but messengers or even avatars of Shub-Niggurath. Therefore, the answer to whether one could keep one as a pet is the same as it would be for a seraph or other divine power: a resounding no.
Assuming one spoke of those created by the Elder Things? Dyer’s translation of the Elder Thing friezes indicates the shoggoths were controlled by some sort of hypnosis; thus, while the precise methods might well differ, they are susceptible to the same pattern of repeated conditioning that is used to domesticate animals. Those shoggoths which rebelled did so long enough ago that they are without doubt irredeemably feral by now; however, without access to the original texts, let alone additional sources, it is impossible to be certain whether the shoggoth rebellion was due to an inherent quality in shoggoths or a result of being abused by their creators; therefore, shoggoths created since the rebellion might—if treated with love—remain loyal.
Of course, shoggoths are large, inquisitive, and covered in an unpleasant slime, so keeping one as a pet would be a constant strain upon both one’s property and one’s sensibilities. However, those who enjoy the company of dogs might well not consider foul slobber and profligate exuberance a barrier. Thus, a certain kind of person of cruder sentiment might keep a shoggoth as a pet.