What is industry standard of how many projects one PM should manage at the same time?
There's no such thing; individual projects vary greatly in the level of effort required to manage the project properly. In addition, each project management framework varies in how much overhead is involved.
Therefore, the correct question is "How many hours per week are really available to our Project Manager for managing projects, and how many hours does each project consume from his pool of available hours?" That question is at least answerable, but only your organization can answer it.
Available Hours and Organizational Overhead
Practically speaking, there's a fixed amount of hours available for each project manager. Whether that's a sustainable 35-40 hours, or whether it's 80+ hours per week until that person burns out, there's always a cap on a person's hours available to manage projects.
From those hours, you need to subtract work-related overhead that isn't really project management or framework overhead. This may include:
- organizational meetings
- time tracking
- responding to non-project emails and meeting requests within the organization
- other miscellanea
In general, you can expect to find that around 4-6 hours per day are the maximum effective hours you can reasonably expect a full-time Project Manager to have left after subtracting this overhead from the pool of available hours.
After subtracting all the organizational overhead from the pool of available hours, you then need to subtract the right amount of framework overhead. This is actually work, but is related to the project management framework rather than the details of the specific project.
For example, around 30% of a project manager's time is consumed by attending framework meetings and generating framework artifacts in Scrum. Other frameworks may differ in the percentages, but most will fall somewhere in the 15-30% range. YMMV.
In addition to everything else, the productivity cost of managing multiple projects is likely to be around 40%. The American Psychological Association says:
[S]hifting between tasks can cost as much as 40 percent of someone's productive time.
In other words, the more projects you put onto a project manager's plate, the fewer projects that person can effectively manage. Again, your mileage will vary.
The Real World
In general, a Scrum Master should manage only a single project at a time. However, that person might be able to handle as many as three if none of the projects are complex or complicated, and task-switching is kept to a minimum.
Even if your numbers are calculated differently, there is still a limit. If you want to run more projects than you have allocatable hours, your organization will need to budget more resources (time, money, and people) to managing those projects. That's rarely what people on a tight budget want to hear, but it's still the truth.