The luminaire (that porcelain thing) is bad news in a closet, whether it has a socket or not. NEC 410.16, "Luminaires in clothes closets". While 410.16 covers what is permitted, 410.16(B) swerves out of its way to outlaw that particular type.
This may not be strictly speaking a clothes closet, but utility closets also get used for things that can catch fire. 410.16 seeks to prevent a particular type of fire, and I have witnessed that type of fire starting (and intervened obviously), and it was a utility closet. More to the point, very good products are available today -- why not use them? So let's use 410.16A as a guidestar, even though it may not be technically a clothes closet.
Where safe to use, the luminaires are fine with or without socket. I use arrays of them for ceiling lighting in basement/shop spaces. They are so cheap that even with steel octagon boxes and EMT conduit, material cost is ~$5 a lamp. It lets me put my glut of obsolete CFLs to good use. Anyway...
410.16A says you can only use one of
- Surface-mount or recessed incandescent or LED luminaires with completely enclosed light sources (i.e. the light emitter is behind a glass or plastic bezel and the luminaire is rated for the heat of the fully enclosed bulb including an incandescent if those exist for the socket type, because somebody will).
- Surface-mounted or recessed fluorescent luminaires (but don't)
- Surface-mounted fluorescent or LED luminaires identified as suitable for installation within the closet storage space (i.e. the manufacturer has specifically listed the fixture for installation in closets.)
410.16C is very particular about location, for every type except 3. I typed in all the gory details, then deleted it because it doesn't matter; it's clear that Code is working pretty hard to herd you into type 3 above.
Figure on this being an LED fixture with integral emitters (no bulbs to change). It's pretty easy to make those cool enough to be safe in a closet. That's the right thing for this job and this location.
Any new closet receptacle needs to be AFCI protected. You might be able to find listed-for-closet LED fixtures with convenience receptacles, but I wouldn't count on it. If your heart's set on a receptacle there, I'd fit a surface conduit adapter box on top of that octagon box, both because it'll give you space for a normal receptacle (with a special cover plate) and also allow you to extend via surface conduit to another location for the luminaire.
You still need to figure out the AFCI thing; you can't put an AFCI+receptacle combo device in an octagon box - it won't fit. So the AFCI will need to go somewhere upstream. Normally you replace the circuit breaker with a breaker+AFCI, because bonus, it also protects the house wiring from problems in the wires or receptacles, which has proven to be so useful they are now required on lots of circuits.
P.S. Here's 410.16 verbatim: