If the sentence is well-written, 'accurately' modifies both 'categorize' and 'draw'. If it were meant to modify 'draw' only, the careful writer would compose it as:
'categorize animals and draw them accurately' or
'draw accurately and categorize animals'
('Animals' remains the object of both verbs in all cases, as you see.)
Because these latter are slightly wordier, the writer of the original may have wished you to discern the logic or make the assumption. Really ambiguous ones do come along, for which you should take your choice and blame the author for any error of judgment.