Does a subpoena that says State of Texas vs. Defendant mean the state is filing charges on him and would it probably happen to me?

Charges are already filed against both of you. That's what the tickets signify. Since these are criminal charges (assault in the form of an unwanted touching with jurisdiction in either a JP or municipal court), the State is the entity that prosecutes the crime. The subpoena is because you and possibly your boyfriend asked for a trial. In response to the trial setting, the state issued a subpoena for the other person because it needs a witness who can testify to prove its case.