This kind of argument is often called a "defense of others" defense which is available in every jurisdiction of which I am aware. Almost every state has a specific description of when this is permitted as part of their criminal code, usually in a general principles section at the beginning, or in the sections pertaining to crimes of violence.
Some details vary from one state to another, but none of the facts presented in this hypothetical really push the envelope in terms of distinctions between one state's law and another.
Most of the differences involve situations when the use of deadly force is allowed. For example, states differ regarding when deadly force allowed to prevent a burglary of your home or business which is in progress, particularly if it is possible to avoid a use of deadly force at all by retreating in a manner that puts you at no one at any significant risk of bodily injury. But, the scenario presented does not appear to involve the use of deadly force (although the definition of "deadly force" can be slippery and lead to some subtle variations in what is permitted from state to state).
Another common nuance of variation between states involves the circumstances under which physical force or deadly physical force is authorized to make a citizens arrest, but this situation is also not implicated by your hypothetical. Once Thug is beat up, the scenario ends without any effort to detain Thug until the police arrive.
Generally speaking, a defense of others defense that justifies a use of force under criminal law will also not give rise to civil liability in a lawsuit for assault and battery as opposed to a criminal prosecution for it.
A Sample Defense Of Others Statute
The pertinent section of the Colorado Revised Statutes (2016), strongly influenced by the language of the Model Penal Code (which never adopted in full by any state but highly influential stylistically in how U.S. criminal codes are drafted) is very typical of the majority rule regarding the defense of others and reads as follows (emphasizing the language relevant to the scenario in the question):
§ 18-1-704. Use of physical force in defense of a person
(1) Except as provided in subsections (2) and (3) of this section, a
person is justified in using physical force upon another person in
order to defend himself or a third person from what he reasonably
believes to be the use or imminent use of unlawful physical force by
that other person, and he may use a degree of force which he
reasonably believes to be necessary for that purpose.
(2) Deadly physical force may be used only if a person reasonably
believes a lesser degree of force is inadequate and: (a) The actor has
reasonable ground to believe, and does believe, that he or another
person is in imminent danger of being killed or of receiving great
bodily injury; or (b) The other person is using or reasonably appears
about to use physical force against an occupant of a dwelling or
business establishment while committing or attempting to commit
burglary as defined in sections 18-4-202 to 18-4-204 ; or (c) The
other person is committing or reasonably appears about to commit
kidnapping as defined in section 18-3-301 or 18-3-302, robbery as
defined in section 18-4-301 or 18-4-302, sexual assault as set forth
in section 18-3-402, or in section 18-3-403 as it existed prior to
July 1, 2000, or assault as defined in sections 18-3-202 and 18-3-203.
(3) Notwithstanding the provisions of subsection (1) of this section,
a person is not justified in using physical force if: (a) With intent
to cause bodily injury or death to another person, he provokes the use
of unlawful physical force by that other person; or (b) He is the
initial aggressor; except that his use of physical force upon another
person under the circumstances is justifiable if he withdraws from the
encounter and effectively communicates to the other person his intent
to do so, but the latter nevertheless continues or threatens the use
of unlawful physical force; or (c) The physical force involved is the
product of a combat by agreement not specifically authorized by law.
(4) In a case in which the defendant is not entitled to a jury
instruction regarding self-defense as an affirmative defense, the
court shall allow the defendant to present evidence, when relevant,
that he or she was acting in self-defense. If the defendant presents
evidence of self-defense, the court shall instruct the jury with a
self-defense law instruction. The court shall instruct the jury that
it may consider the evidence of self-defense in determining whether
the defendant acted recklessly, with extreme indifference, or in a
criminally negligent manner. However, the self-defense law instruction
shall not be an affirmative defense instruction and the prosecuting
attorney shall not have the burden of disproving self-defense. This
section shall not apply to strict liability crimes.
Analysis of the Hypothetical Facts
Is Paul guilty of assault and battery for attacking Thug, or would
self-defense or some other standard defense apply?
Paul is probably not guilty of assault and battery for attacking Thug because a defense of others defense justifies his actions.
Paul is using physical force to defend a third person (and presumably himself as well once once he is involved in the fray) from what he reasonably believes to be the use of unlawful physical force by Thug, and generally, under those circumstances Paul may use a degree of force which Paul reasonably believes to be necessary for that purpose.
Paul is most vulnerable in this scenario on the question of whether he reasonably believed that the degree of force he used was reasonably necessary for the purpose of defending Emily and himself from Thug's unlawful use of physical force, or whether he went further than what was reasonably necessary for a few minutes that left Thug "incapacitated and thoroughly bloodied".
For example, suppose that Paul had already caused Thug to try to flee the scene after the first minute at which point Thug was slightly bruised and afraid of Paul, but was not at all incapacitated. But, suppose that despite this fact that Paul, awash with adrenaline from the fight, continued to pummel Thug for a couple more minutes as Thug repeatedly tried to flee the scene only to be dragged back by Paul. Suppose that Paul continued the fight after it was no longer necessary because Paul wanted to punish Thug for his mistreatment of Emily and to discourage other people from trying to attack Emily in the future, even after Paul knew that there was already no real risk that Thug himself would continue to use unlawful physical force against Emily no or in the near future.
Under these circumstances, which aren't inconsistent with the hypothetical facts, if this extra couple of minutes caused Thug to be much more badly injured than he otherwise would have been if Thug had been allowed to flee after the first minute, then Paul could still be guilty of assault and battery, even though his actions were legally justified for the first minute of the fight.
There is no indication that either Paul or Emily provoked the attack on Emily, or that either Paul or Emily was the initial aggressor, or that this was actually a pre-agreed dueling situation. So each of these circumstances which would be exceptions to the general rule that the defense of others is justified do not apply to this case.
There is likewise no indication that Paul used "deadly force" as opposed to mere "physical force" in handling the situation. It is likely that Paul would have committed a crime if he had shot and killed Thug instead of beating him up, if Paul knew perfectly well at the time that he was capable of beating up Thug and making Thug go away for good as a result without resorting to a firearm, unless the attack on Emily was severe enough for Paul to reasonably believe that the attack was putting Emily at a real risk of serious bodily injury.