Melachim 4:4: Similarly [the king] may take from all the borders of Israel wives and pilagshim: wives with kesubah and kiddushin, and pilagshim without kesubah and without kiddushin, just with yichud (does he mean seclusion, or her being chosen or designated even before the seclusion with the king ever took place? In other words, is he using Yichud in the ishus meaning or not?) alone does he acquire her and she is permitted to him. But a hedyot is forbidden to have a pilegesh except for an amah ha'ivriah after yiud.
The Rambam's opinion is very clear that a pilegesh is only allowed for a king, and a commoner does not have a pilegesh except for ama ha'ivriah after yiud. There is no concept of common-law marriage as the definition of pilegesh. He holds Yiud obligates a man to provide for this woman as his wife, but this obligation did not take place through kiddushin. So the result is the same as marriage, but the mechanism is different - amah ha'ivriah with yiud versus regular kiddushin. In fact, the Rambam does not list the laws of yiud in Hilchos Ishus. They are codified in the fourth chapter of Hilchos Avadim. But some mechanism is needed to become "married," so a common-law relationship would not even be considered a pilegesh.
הלכות אישות פרק א
[א] קודם מתן תורה, היה אדם פוגע אישה בשוק--אם רצה הוא והיא לישא אותה--מכניסה לביתו ובועלה בינו לבין עצמו, ותהיה לו לאישה. כיון שניתנה תורה, נצטוו ישראל שאם ירצה האיש לישא אישה--יקנה אותה תחילה בפני עדים, ואחר כך תהיה לו לאישה: שנאמר "כי ייקח איש, אישה; ובא אליה" (דברים כב,יג).
[ד] קודם מתן תורה, היה אדם פוגע אישה בשוק--אם רצה הוא והיא--נותן לה שכרה, ובועל אותה על אם הדרך והולך לו; וזו היא הנקראת קדשה. משניתנה התורה, נאסרה הקדשה--שנאמר "לא תהיה קדשה, מבנות ישראל" (דברים כג,יח); לפיכך כל הבועל אישה לשם זנות, בלא קידושין--לוקה מן התורה, מפני שבעל קדשה.
We see what marriage is, we see what a kedeisha (commonly translated as prostitute) is - she will have a sexual relationship with anyone. Note that he does not discuss an intermediate case of common-law marriage - where she has a sexual relationship only with one person, but without kiddushin. Rav Ahron zt"l said that there are two parts to full ishus - the ishus of a non-Jew, which is the ishus of before matan Torah, and the second part is chuppa with eidim. I don't remember the context in which he said this, or what question he was answering with this.
Ravad 1:4 says that a common law marriage - where one woman had an exclusive relationship with one man is not assur midioraissa, and that is called a pilegesh.
Maggid Mishna 1:4 says the Rambam's opinion is that pilagshim have kiddushin but no kesubah. The Kesef Mishna (1:4) asks against the Maggid Mishna that the Rambam in Hilchos Melachim (4:4) explicitly says pilagshim have neither kiddushin or kesubah. Kesef Mishna also points out hw the Ramban in the teshuva also misrepresents the shitas Harambam, also by apparently not having the text of Hilchos Melachim as we do.
Some, including Ramban (Teshuva 284) ask against Rambam why a king can have pilegesh and a hedyot can't, that there is no source for it. We see, however, that Rabbeinu Yonah (Shaarei Teshuva 3:94) says "that a king, whom everyone fears, and no one will commit adultery with her." So that is reason enough to distinguish a king from a commoner. Margalios Hayam, Sanhedrin 21a paragraph 10 brings a similar Maharit: since a pilegesh of a king is forbidden to everyone else, she is not a kedeisha. Note: Many, including Margalios Hayam, ask against Rambam that Gidon was allowed to have a pilegesh because he was shofet - interesting source: see Frankel edition, Rambam Hil. Terumos 1:2, and Semag that a shofet is like a king for kibush rabim, hence a shofet may have a pilegesh (See Rav Ahron Soloveichik, Perach Mateh Aharon Ahava p. 159: A shofet cannot effect kibush rabim because he is only the leader over individuals, whereas a melech can, because he is the leader over all of Yisrael as one unit, not as many individuals. He published this piece in the 1960s in a Beis Yitzchok (RIETS) before the Frankel edition of Rambam was published and showed from manuscripts that a shofet does effect kibbush rabim; I never saw him use the Frankel Rambam even in the 1990s.) Note that this would not answer Ramban's question that we saw that the Pilegesh B'givah incident was where a commoner had a pilegesh. Rambam would answer simply that in those cases, a pilegesh was a woman who was originally an amah ha'ivriah, with yiud.