How can the organic mercury compound Thiomersal be not dangerously toxic?
Thiomersal is broken down into two compounds, thiosalicylate and ethylmercury (Ensink 2015).  The thiosalicylate it is relatively non-toxic, and there is even some evidence that this compound could counteract some of the effects of mercury poisoining (Asadi et al 2010), though the mechanism of this inhibition has yet to be elucidated.
Although they have similar names, methylmercury and ethylmercury have very different levels of toxicity.  Methylmercury has long been known to be highly toxic, producing high levels of cell death in the brain, particularly in the cerebellum and the visual cortex (Ceccatelli et al 2010).
Unlike methylmercury, ethylmercury appears unable to cross the blood-brain barrier as fast as it is eliminated in the stool.  Consequently, small quantities of ethylmercury have been shown to be non-toxic, resulting in the relative non-toxicity of thiomersal (Pichichero et al 2002).

Sources:

Asadi S, Zhang B, Weng Z, Angelidou A, Kempuraj D, Alysandratos KD, Theoharides TC. 2010. Luteolin and thiosalicylate inhibit HgCl(2) and thimerosal-induced VEGF release from human mast cells. Int J Immunopathol Pharmacol 23(4): 1015-1020.

Ceccatelli S, Daré E, Moors M. 2010. Methylmercury-induced neurotoxicity and apoptosis. Chemico-Biological Interactions 188(2): 301-308. doi:10.1016/j.cbi.2010.04.007

Ensink E. 2015. Preservative or poison? The science behind Thimerosal. The Substrate (online): http://substrate.asbmb.org/2015/05/15/preservative-or-poison-the-science-behind-thimerosal/

Pichichero ME, Cernichiari E, Lopreiato J, Treanor J. 2002. Mercury concentrations and metabolism in infants receiving vaccines containing thiomersal: a descriptive study. The Lanclet 360(9347): 1737-1741. doi:10.1016/S0140-6736(02)11682-5