There is such a thing as SMS Spoofing, and it works similar to the way phone number spoofing occurs.
If the problem isn't due to an app which holds permissions to be able to make calls and send SMS (Please check this! The permissions will tell you what could cost money!), it's probably spoofing. Unfortunately, not much you can do about it. Telephone companies like Verizon have to hold a responsibility to secure down mobile networks and PSTN to ensure spoofing cannot occur.
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