"Bank" is a good example of a homonym (same spelling and pronunciation but different meaning), a homograph (same spelling but different meaning) and a homophone (same pronunciation). I can think of at least three different meanings of the word in its noun form - a place where financial transactions are made, a mound or heap, and a set of matched or similar objects.
When referring to a place for depositing money or other property, the word is derived from the French word "banque" which originally meant a table or bench. When referring to a mound or heap, "bank" has direct Scandinavian origins. Although it seems that etymologists trace the French "banque" to the Italian "banco" which also refers to a bench, I wonder if the "banque" didn't enter the English language via the Normans who inhabited the part of France invaded by the Vikings and who in turn invaded England. It's also possible that the Scandinavian sense of the word entered England directly courtesy of the Vikings who also occupied much of Britain.
The California Secretary of State won't allow articles of incorporation to be filed that include the word "bank" unless the approval of the Commissioner of Financial Institutions (now known as the Commissioner, Department of Business Oversight) is attached. Cal. Corp. Code Section 201(a). But what if a business wants to use the word in a non-financial sense? For example, a restaurant might want to incorporate under the name "Sacramento River Bank Sushi Co." While it seems very unlikely that anyone would try open a checking account at a corporation with that name, the Commissioner of Financial Institutions in a 1998 letter wrote:
However, it appears to us that the statute leaves no room for interpretation as to names in which the word "bank" appears. It expressly provides that articles setting forth a name containing the word "bank" not be filed unless the Commissioner has issued a certificate of approval. Accordingly, it does not appear to us that the Secretary of State has authority to file articles of incorporation containing terms such as "food bank," "blood bank," or "river bank" unless the articles are accompanied by a certificate of approval from the Commissioner.
The rules become less logical when applied to foreign corporations. A foreign corporation (other than a foreign association) that is not transacting intrastate business may register its name with the Secretary of State with one proviso - the name must be a name available under Section 201. Cal. Corp. Code § 2101(a). This puts a foreign corporation in the same position as a business to be incorporated under the General Corporation Law.
Foreign corporations that are transacting intrastate business must, of course, must qualify with the Secretary of State. Cal. Corp. Code § 2105. These corporations may not qualify if they have a name prohibited by subdvision (b) of Section 201. Cal. Corp. Code § 2106(b). The statute doesn't mention subdivision (a) (which as noted above prohibits the inclusion of the word "bank"). Thus, a foreign corporation transacting intrastate business may qualify with a name that includes "bank" while a foreign corporation not transacting intrastate business may not register that name and a business may not incorporate with that name under the General Corporation Law. If you're looking for a rational explanation for this set of rules, I have none.
What about other (mis)spellings of "bank" such as "banc" or "banque"? Subdivision (a) of Section 201 doesn't mention these alternative spellings and thus they don't strictly fall under the prohibition. However, subdivision (b) of Section 201 broadly prohibits names that are likely to mislead the public. Thus, a name that suggests that a company is a financial institution, but isn't, may nonetheless run afoul of subdivision (b).