Can I claim that a Company is my customer and show their name and logo on my site?

Your mention of a company's trademark to identify them as a customer isn't the kind the use the trademark laws are designed to prevent, because you're not a competitor of the company and not using the mark as a trademark.

But I don't think it's accurate to represent that a company is your customer when it's the employee (who probably doesn't have the authority to act for the company or to authorize the use of the company's logo) who's really your customer. Just because they use their company's email to sign up for your product/service doesn't make the company your customer.

You can identify the employee as an employee of the company without violating the trademark law, but you're going to want to get the employee's consent, especially if you use their full name and title at the company.

Disclaimer: Please note that this answer does not constitute legal advice, and should not be relied on, since each state has different laws, each situation is fact specific, and it is impossible to evaluate a legal problem without a comprehensive consultation and review of all the facts and documents at issue. This answer does not create an attorney-client relationship.