This is nothing "accidental" on my part, e.g. any of the strange and wonderful things that a phone might try to do while rolling around in your pocket. :-) After a lengthy (chat ...) conversation with Verizon, they suggested that I might wish to replace the SIM-card in this phone – which can be done for free at any "corporate" store – just in case the phone might somehow have been "hijacked" or "cloned." Although I consider this possibility to be very unlikely (I live in the country ...), I plan to do it. They told me that they did not have access to the thread itself – and I must take their word on this, I suppose – so as to be able to determine from their end what might actually have happened. They took the transcript of my 'chat' and agreed to open and escalate an internal incident report within Verizon.
Since this incident involved "both an iPhone and an Android," hence an SMS conversation and not iMessage, I'm not really sure at this point if there is really anything else to be done. Nor anything in particular that "Apple" could do. So, I must consider this to be "closed."
Thank you, all, for your prompt and valuable contributions. One thing that this exchange has told me is that at no point did anyone step up and say, "Hey! This happened to me, too!" Neither did Verizon indicate such.