TLDR: No wrongdoing, just get on with the job unless someone calls you out (which is extremely implausible).
There is no wrongdoing on the job applicant's part. Sending a job application to a recruiter does not prohibit you from applying to other job postings found through other channels.
It is not uncommon for job applicants to apply to hundreds of job postings, through dozens of different channels, such as employee referrals, recruitment agencies, job search portals, directly to company's website, walk-in interviews, and so on. It is hard enough to remember all the job postings you have applied to, leave alone being able to figure out similarities between various job postings.
That aside, there are too many uncertainties, beyond the control of the job applicant, such as:
- The most obvious one: the job postings only look similar, but they could well be from two different companies who happen to have the similar requirements.
- The recruitment agency may have "pre-rejected" the application, based on their (flawed) understanding of the company's requirement. The agencies typically get rated on the quality of applications they send to companies, so it is good for their business to reject "junk" applications on their own.
- The company may have "withdrawn" the requirement from the agency before the agency got to sending the application.
- The application through the agency was rejected by Hiring Manager A, while the direct application was accepted by Hiring Manager B.
and so on. The applicant couldn't know what transpired behind the curtain, and he couldn't do anything about it anyway, so it is best he doesn't worry about it.
Depending on what actually happened behind the curtain, there could be a case of wrongdoing between the company and the agency. For instance, one scenario is:
- Company sends the requirement to the agency (or agencies), because they thought few candidates would be interested and they couldn't be bothered looking for needles in the haystack.
- Recruitment agency uses all its channels to get interested candidates, and sends applications to the company.
- Looking at the volume of applications, company says, "Hey, maybe this isn't as hard as we thought. Why don't we just deal with it ourselves?"
- Company ditches the agency, and now makes a posting on its website, hoping that most of the interested candidates would get there and apply.
Even so, there is nothing the candidate needs to do (or could do, even if he wanted to).