That's not quite correct. A lesser included offense (LIO) must have a strict subset of elements of the greater offense and/or a less culpable mens rea. For example, possession of cocaine is an LIO of possession of cocaine with intent to distribute; homicide by reckless and wanton disregard of human life (felony murder/third degree murder) is an LIO of homicide by malice aforethought (premeditated murder/capital murder/first degree murder). In other words, LIOs are those offenses which are
logically necessary to commit the larger offense. Charging both an offense and its LIOs is what, in our jurisdiction, we call multiplicity.
What this discussion touches on is not multiplicity but unreasonable multiplication of charges (at least in my jurisdiction we call it that). The test here is not the logical question of whether one necessitates the other by definition; it's the factual question of whether the offenses are part of substantially the same transaction. For example, a prosecutor might charge perjury, fraud, and larceny, wrt someone swearing to some materially false fact and fraudulently obtaining someone else's property with intent to permanently deprive them of it. These three offenses are not subsets of one another, but are closely related; it isn't logically necessary to commit perjury to commit fraud, in the abstract, but in a particular case that might be the mechanism by which one commits fraud. In my jurisdiction, UMC charges can stay on the charge sheet (perhaps the jury finds guilt on one but not all?) but, if convicted of more than one, they are merged for sentencing.