Alternative medicine is medicine that is outside the standard of care and typically has no strong evidence for effectiveness beyond placebo effects. If there is sufficient scientific evidence that a treatment is effective, it becomes part of the standard of care and is no longer alternative medicine.
Alternative medicine does not mean "using 'natural' ingredients" or anything like that, because "natural" doesn't have any relationship to what is safe or effective: naturally occurring compounds can be poisons just as effectively as synthetic ones can be, chemistry doesn't care about the history of the atoms included in a molecule.
Lithium used as a medicine has always been based on experiments that showed effectiveness (even if mechanisms are/were poorly understood). Therefore, no, it is not an alternative medicine when used to treat anything for which there is evidence of effectiveness.
Lithium has also been used as patent medicine - in that context it could be thought of as "alternative" in that marketing claims were made that were not based on scientific evidence.
Lithium was not originally regulated, nor was anything. Regulation of anything is a relatively recent phenomenon, with the US FDA for example only founded in the early 1900s. Not everything is regulated immediately, new regulations are continually added as evidence for harm is discovered. Pharmaceuticals are a bit of a special case in that they must be "approved" rather than simply absent from a "harmful" list, but their regulation is related to marketing them as medical drugs, not based on their production or consumption.
Marmol, F. (2008). Lithium: bipolar disorder and neurodegenerative diseases Possible cellular mechanisms of the therapeutic effects of lithium. Progress in Neuro-Psychopharmacology and Biological Psychiatry, 32(8), 1761-1771.
Shorter, E. (2009). The history of lithium therapy. Bipolar disorders, 11, 4-9.