How can I convert a 240V dryer outlet to two branch circuits?
Yes, it is legit to have a “shared neutral” circuit, to be more specific, a “multiwire branch circuit” or MWBC. The Code forces you to engage in certain best practices, because even if you get it perfect, it's very easy for the next guy to inadvertently create a hazard. 

First, 2011 code absolutely requires a 20A ganged breaker (or approved handle ties).  Aside from the reason you recognize, it also assures you are on opposite poles of the 240v, which avoids overloading the neutral.

All neutrals on an MWBC must be “pigtailed” - google it, i.e. you are not allowed to daisychain through outlets by using the extra 2 terminals.  (Well, OK, you can daisychain the “hot”.)  This is to prevent “lost neutral” damage downstream if an outlet is removed or has a problem. 

Exception: Your GFCI outlet.  By nature, they require a dedicated separate neutral to any downstream outlets, so naturally you must daisychain both hot and neutral for those.  To make that clear, mark that neutral in a distinctive way so the next guy doesn't grab it thinking it's the MWBC neutral.

To answer your followup, opposite poles matter.  While your homerun is 40A rated to the old plug, you certainly are extending the MWBC beyond it, and using 20A wire (e.g. 12 gauge) to do so.  In the future, others will add to that MWBC. What happens when the next homeowner runs the other "hot" out to the workbench? Due to time/billing/skill limitations the "next guy" can't reverse engineer the full scope of every circuit they touch - they must be able to assume it's correct with no hidden gotchas, which is a big part of why there's a code.  That is especially important with MWBCs because they are so touchy. 

If the restrictions or safety margin of MWBCs bother you, slap a sub-panel at the site of the old plug, and run simple circuits off it.