An Etymological Dog Kennel, or Dog Eat Dog: Icelandic setja upp við dogg, Engl. to lie doggo, Engl. dog, and Engl. it’s raining cats and dogs - Liberman, Anatoly (Scandinavistica Vilnensis, 2019)
There is a rare slangy British English phrase to lie doggo “to lie hid”. The earliest known example is dated in the OED to 1882. Doggo looks like dog + o (with -o, as in weirdo, typo, and so forth), but a formation consisting of an animal name followed by the suffix -o would have no analogs. Some light on the origin of lie doggo may fall from the Modern Icelandic idiom sitja upp við dogg ‘to sit or half-lie, supporting oneself with elbows’. Doggur, known from texts since the eighteenth century, occurs with several other verbs. Also, sitja eins og doggur ‘sit motionless, look distraught’ and vera eins og doggur ‘to be motionless’ exist. Doggur has nothing to do with dogs, because the Scandinavian word for “dog” is hund-. The origin of the English noun dog is obscure, but, contrary to the almost universal opinion, the word is not totally isolated. In some German dialects, the diminutive forms dodel, döggel, and the similar-sounding tiggel teckel occur. Perhaps dog and its continental look-alikes were originally baby words. The same sound complexes as above sometimes mean ‘a cylindrical object’ (such are Icelandic doggur and Middle High German tocke). Two of the basic meanings of those words were probably ‘round stick; doll’. Although the evidence is late, we can risk suggesting that lie doggo also contains the name of some device that was current not too long ago in the European itinerant handymen’s lingua franca. The overall image looks nearly the same as in the phrase dead as a doornail. In English, folk etymology connected doggo with the animal name and misled even professional lexicographers and etymologists. Finally, of some relevance is the English idiom it rains cats and dogs, whose forgotten earliest form was it rains cats and dogs and pitchforks with their points downwards. Apparently, the original idea was that a downpour of sharp objects fell to the ground.