it was a late application.
It means it went into the bin "if we really need to fill the position, and we have no other candidates, let's consider this applicant, too"
they communicated to me that they did not have time to review my case
They did not review your case.
and they wanted to keep it for the next round.
They keep it just in case they don't fill the future position with the future applications, and with the future late application. Plus there may be some strange data protection law, where they have to log through long processes to manually delete your application, so it is easier for them to keep it open in their records until automatic deletion incurs.
explicitly mentioned that this is not a negative result
In fact it is not any result, they said "we did not review your case".
So, going to your question(s):
- Should I contact him again as the results will be released several months from now?
Not useful to contact him about the results, very useful to contact him proactively to write a proposal to fund your research with him/her indipendent of the PhD selection process.
For that specific PhD application? Yes. Wait next year application round and go through that.
- is it acceptable to apply to other universities, in the meantime, with the same research proposal?
Yes. But be smart: try to align your proposal with the state-of-art of the focus of the department research . Do it without taking your proposal as a founding stone, but as a raft with which you sail through the topic. You may have some pleasant but uncomfortable surprise, after feeling drowned in the subject (this department is completely wrong?!) you will realize you can swim in the topic of your choice.
 example: if you are interested in research on water cycle and water movement, you may be interested in seeing how rainwater infiltrates the soil and ends up in agricultural fields. However, it would be not wise to propose this topic in the context of "rainfall in the Tibetan plateau" to a geophysics department that is switching from "climate research" to "industrial applications" ...