Can we fetch the depth of taxonomy from term load
You can use TermStorage loadTree function. It will load the whole tree and the depths of each term will be avaiable. 
A much more cleaner / efficient way can be to use the "taxonomy_term_hierarchy" table, (database service, select..). You query over and over again till the "parent" (or current examined) term parent is 0. 


If you really feel you are being lost, take a look on this thread.