Can you give an app the same name as an already existing company uses for their website (same industry)

The answer to your question is fact sensitive. What goods/services do they provide on their website? Is that business name actively registered as a Trademark with the USPTO? If so, under which filing class(es)? How do you plan to use your app? The USPTO has 45 classes, or categories, that cover all goods and services. The scope of protection of a federal trademark registration is only applicable to the specific international class(es) and the goods and services that you designate and pay for in your trademark application. For example: a Retail Pet Store falls under Class 35 (Services), Pet Grooming falls under Category 44 (Services), and Downloaded Computer Application Software for Mobile Phones fall under Class 9 (Goods). Beyond the general identification of these classes, are a description of each individual good or service that the Business wants their trademark to apply to. The takeaway here is that, it is possible for two identical trademarks to coexist when they’re used for two unrelated classes and sometimes even when they are used for the same class if there is no likelihood of confusion. Consulting with a Trademark Attorney is critical here to help you address these questions and help you identify any legal barriers to your company's use of the mark.

As my colleague commented, if you create a likelihood of confusion, that's trademark infringement, since consumers won't know if you're them, or they're you, or if you've got a formal business affiliation with them, etc., which could get you unwanted attention from this website's lawyers. You're best off choosing an original and distinct name that no one else has. And since your business name/trademark will likely be your company's most valuable asset, it's wise to seek professional help selecting and acquiring trademark rights. About 1/2 of federal USPTO trademark applications get rejected, so lots of applicants must think it looks like an easy task. It's not.

You naming your app is creating a trademark. The test for trademark infringement is likelihood of confusion in the marketplace by your customers or the customers of the website. If, as you say, you are going to be in the same industry, will your customers go to the website thinking they were going to reach your app? Or, will the customers of the owner of the website download your app because they think it is affiliated with the website? If either answer is yes, there is trademark infringement. Think about it: would you want someone to start a website in the same space your app is in using your trademark? Probably not. To be sure of the potential landmines you might coming close to, it would be best to get the opinion of an intellectual property attorney. Find one and discuss your issues in private.