This appears to be classic "fisheye", a defect caused by contamination of the surface by silicone oil. In a furniture context the chief source of this is modern furniture polishes, although there are numerous other possible sources these days. Be wary of spray polishes in general and anything that promises a wet-look shine in particular people!
This is one of the worst cases I've seen photos of as it seems to involve all the visible surfaces to almost the same level (generally the effect isn't as pronounced and/or is more localised).
Given this piece has already undergone certain expected stages in a refinish — stripping of the previous finish (generally followed by some amount of rinsing/wiping off of residues with an organic solvent or water) and then at least a little sanding — it begins to illustrate how difficult it is to remove this contamination.
Standard finishing and refinishing processes must be assumed to never be effective, you have to specifically combat it.
There are various methods used to try to remove the contamination, or seal it in. These include, but are not limited to, the following:
cleaning with ammonia1;
cleaning with sodium hydroxide2;
cleaning with TSP;
rinsing of the surface with hexane3;
rinsing of the surface with xylene4;
sealing the surface with light coats of shellac.
Just to highlight how pernicious the contamination can be, you can use a combination of the cleaning steps, being scrupulous about how you use your wiping rags or paper towels5, you can do multiple rounds, AND then apply shellac and still find you were unsuccessful o_O
This unfortunately leads us to the one technique known to almost always work.
Deliberately add silicone oil
Yes, you actually add silicone oil, or a product containing it such as "fisheye eliminator", into the new finish. How this works is that it lowers the surface tension of the new finish until it flows evenly over the contamination.
I say unfortunately because although it's effective A) it of course contaminates all the application tools, which may be very difficult or impossible to subsequently clean, much more importantly B) it risks contamination of the refinisher's finishing area and potentially their entire premises6 C) pieces treated this way will forever need to be finished the same way, and some refinishers may actually not want to touch it as a result.
This, especially in combination with some careful spray methods, is almost guaranteed to successfully combat the problem.
Further reading (not a comprehensive overview):
Silicone Contamination on Highland Woodworking.
Fish Eye & Silicone – or, the Problem with Furniture Polish on Popular Woodworking.
Refinishing a Silicone-Contaminated Antique on Woodweb (be sure to compare and contrast some of the feedback here with Flexner's advice).
How to Prevent and Eliminate Fisheyes on Fine Woodworking (note: partially behind paywall).
Also see the disclaimer suggested by General Finishes on this page. It starts with the realistic but depressing:
There are no guarantees with silicone - it is almost impossible to remove.
1 This is a risky method when it comes to wood as many wood species are affected by stronger alkalis and change colour.
2 AKA caustic soda or lye. The caution related to ammonia goes double here, and in addition caustic soda solutions are extremely hazardous and pose quite a risk to the user as well.
3 This is more common in non-woodworking contexts because hexane is often used as a cleaning liquid in industrial or technical applications.
4 Xylene is a more common solvent in a woodworking context; this is basically the same process as the preceding one just using an alternate solvent.
5 A step that many get wrong; the correct procedure is to wipe once and then discard, to do the utmost to prevent cross-contamination of an unaffected area from the affected areas.
6 No this is not hyperbole.