I think that these, believe it or not, convey distinct meanings and aren’t redundant. They use slightly different senses of the words.
For example:
“It’s possibly possible for humans to colonize Mars/create sentient AI.”
“It’s possibly possible for me to lift 400 pounds if I became a bodybuilder.”
“It’s possibly possible for Earth to combine with another planet”.
That is, it may be possible for humans to colonize Mars or create sentient AIs, or for you to lift 400 pounds, but we don’t know yet. (Without making empirical claims, let’s assume for sake of example that we know this to be true.) The first meaning of possibility relates to chance/probability whereas the second relates to ability/capability/feasibility. Another way to express it more clearly is “It’s potentially possible/there’s a chance that it’s possible/capable of happening or being done”. To just say “It’s possible” might convey the same meaning, but risks confusing/conflating it with saying that it’s definitely or even currently possible, rather than hypothetically (either physically/nomologically or just logically) possible in the sense of not currently being able to rule it out, falsify/disprove it or prove it either way (but having reason to think it could happen in some scenario).
“It’s not necessarily necessary for you to wear a tie to the meeting, they might not care.”
“It’s not necessarily necessary to use antibiotics to treat the infection”.
It may be necessary (in achieving a stipulated goal/outcome, such as success in getting a job or promotion, or tackling an infection) for some conditions or event/actions to be true/occur like wearing a tie or using antibiotics, but not definitely (we can’t guarantee it would be required). First necessary relates to degree of certainty/confidence/conviction or accuracy/completeness of what you’re saying, second relates to its requirement for serving a purpose (whether or not the purpose is itself necessary/required in any way). One way to say it might be “It’s not definitely/absolutely/guaranteed to be required/essential”.
How speak?
I think it is redundant. The word “possible” is already vague. Adding “possibly” just adds vagueness to something that’s already vague. Another way to look at it:
“It’s possible.” == It can happen. Doesn’t necessarily mean it will, just that is could/can.
“It’s possibly possible.” == means the same thing.