A popular restaraunt here in Yerushalayim has a small idol on its cash register. Every paying customer will see it. The staff will answer your questions about it, and recommend reading. Its' certified kosher. Is this a mistake? Or is a restaraunt allowed to have an idol, but not work on shabbat?
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In Israel the courts will force the local Rabbanut to give certification under such circumstances.
So the Kosher certification has no bearing on what is acceptable under normal circumstances. If, in addition to the standard Rabbanut certification there were also a private certification, that would be a question for that private certification. They may not even be aware of it.
"Kosher" is a pretty vague term. It can mean that it may be eaten according to Jewish law, or it can mean fit for any given purpose.
In the context of restaurants, kosher usually means that the food served may be eaten, but can also refer to other aspects of the institution. It depends mainly on the hechsher (certification) - some care about more than others.
When it comes to Shabbat-observance, there are additional problems with regard to the food itself. For example, wine poured by a non-Shabbat-observant person, Jewish or not, may not be drunk.
So having an idol in the store doesn't affect the acceptability of the food (unless someone actually worships it, which could cause problems similar to Shabbat). Some hechsherim will offer their certification to any store with kosher food; others will require complete compliance with halakha (which forbids owning or having an idol in one's possession).