The below does not constitute legal advice, does not form an attorney-client relationship, and should not be relied upon to take or refrain from taking any action.
The short answer is yes, it is possible for more than one company to have trademark rights in the same word or phrase in different industries. Examples of this include Delta (for faucets and airlines) and Pioneer (for electronics and agricultural seeds). The question in these kinds of circumstances is often whether the goods or services offered by the two companies are sufficiently related such that a consumer would be likely to think a company providing the first product or service would be reasonably likely to provide the second product or service also. To take one of my examples above, it's unlikely that someone would think a seed company would also produce electronic equipment, so even though both companies use the same trademark, consumer confusion is unlikely and the two companies can coexist.
Turning to the specifics of your question, a first point is it doesn't matter the name of their company, just what term they are using as a trademark. So, if they sell their food product under the name "Sunny Florida," it doesn't matter from a trademark perspective if their actual company name is "Cloudy Maine." To determine whether there will be a potential conflict, you have to ask the question I indicated above: would it be reasonably likely that a customer would think the company providing the "Sunny Florida" food product would also be affiliated with your business? Here there's not enough information to know for sure, but I would at least be cautious. This is because it would probably be reasonable for a customer to expect that a food company would have a website, and it would be reasonable for them to add the word "online" to their trademark. So, even if consumers wouldn't be confused, it may be that the Sunny Florida food people might object.
You should discuss the specifics of your case with a trademark attorney who can give you a better idea of the potential risks after getting more detail.
Kurt's answer was right on.
I would add that if a trademark is merely descriptive of a product or its characteristics, the owner might not have acquired legal rights to exclude others from its use, even on similar products. If the other company's trademark was registered on the Supplemental Register, rather than the Principal Register, that indicates that the trademark office considered the term at the time to be descriptive of the goods, but potentially capable of becoming distinctive as to that company's product through ongoing use of the trademark and its association by the public with the particular product.
Similar considerations would apply with regard to your own trademark. Be sure to ask about this when you consult an attorney experienced in working with trademarks.
You have received good advice.
I just want to point out that there is no way to guarantee that you won't hit a road block in the application process. Responding to an Office Action based on likelihood of confusion is not something a layperson should take on. Please speak with a trademark attorney (many charge flat fees) before proceeding.