From your experience, is it some kind of sign when handling editor of your manuscript (which is under review) sends you a review invitation for other manuscript?
Yes, it's a sign that they don't think you're a complete crank. That may not sound like progress, but it is a good thing nevertheless. It doesn't mean anything else about the handling of your own paper.
And remember, part of the art of getting published is keeping on the good side of editors.
No, it's no sign at all other than that the handling editor thinks you are qualified to review the article you were invited to review.
Apart from the fact that you are seen as a person qualified for providing a review, this CAN (half-jokingly) be the sign of desperation of laziness from the side of an editor. Knowing that you are currently enjoying the services of the journal you are perhaps more likely to accept doing the journal a service as a reviewer. The situation you mention has happened to me quite a few times where a rush of requests from a specific journal has followed after submitting a manuscript to that journal. Of course an eaqually or more likely explanation is that the editor is running thin on good contacts that have not been asked already and now finds a new fresh name that can cover topics that correspond to manuscripts of the editor's desk.
In the end, you will never know, but as alrady stated elsewhere, you are deemed as a person with a standing or reputation that qualifies you to provide valuable input as a reviewer and that is all that you need to know.