@Erudin has it. The fitting in your picture is just a variant of the plastic/metal cup that you'd find on a, say, Shimano UN52. There should be a similar one on the other end. Work out which is the "adjustable" cup (the one that has a taper and tacks up the slack on fitting) and undo that first, then undo the "fixed" cup (usually, umm, fixed to the body of the BB). Likely causes of the lateral play and clicking are (a) the adjustable cup has worked loose, so tightening it up with fix it or (more likely) (b) the bearings in the BB are worn out. Remove the whole thing and fit a Shimano UN5x for about fifteen quid, where x is 2,3 or 4 depending on what's in stock.
And the tool, again as Erudin says fits all six splines. I couldn't find mine, as the Teetering Piles of Cr@p in the garage have eaten it.