What is the material and how to seal it?
The painted coating (it looks to be at least the most recent coat of paint and an additional layer below, of either paint or primer) has obviously failed. The cause of the failure is most likely inadequate surface preparation prior to application, exacerbated by high-humidity conditions.
It is very unlikely that the ceiling plaster itself is &quot;off-gassing&quot; or that it is, in and of itself, the cause of any odor. What is likely is that the odor is mildew growth, of which there are some tell-tale signs of along the edge of the wall.
You need to:

strip all unsound material
chemically kill all mildew (there are various methods; I use dilute bleach
sand the entire substrate (while also patching any gaps and defects such as the linear crack where the wall meets the ceiling)
clean/wash the entire area
properly prime and paint using high quality paint products and professional grade applicators

NOTE- a special bonding primer (e.g. shellac) is not necessary if you prep the job as described.
NOTE- a paint with a sheen, such as gloss or semi-gloss, would tend to resist problems associated with moisture better than flat paints.
NOTE- pro-grade applicators and high-quality paints result in a better looking, better performing, and easier job.
NOTE- for longevity and bio-growth prevention, you should ensure there is adequate ventilation. Now is the time to consider upgrading your exhaust fan.