"-P" is a suffix that a person can attach to the end of someone's name to indicate closeness, familiarity, and affection. These suffixes are all called "honorifics" in English but are called 「呼び方」 (yobikata) (or far less commonly, 「呼ばせ方」 [yobasekata]), which mean "how to call (someone/something)" in Japanese. "-P" is one of the examples like the far more common "-chan" that do not signify respect and honor (like the standard "-san" or the more formal "-sama"), so these affectionate suffix kinds of yobikata are called 「愛称」(aisho), which means "affectionate nickname." (In Japanese, these suffixes are not attached with a dash; the name runs right into the suffix like 「美奈P」[MinaP] or Makoto's「まこちゃん」[Makochan]. The dash when writing in romaji sets off a suffix from a name for non-native Japanese language learners to be able to identify where the name or abbreviated nickname ends and the suffix begins.)
Japanese people use yobikata almost all the time to everyone they know, from family members up to the highest elite person (a person trying to be cutesy can even attach one to his/her own name, which Minako does in certain moments). To avoid attaching a suffix to someone's name is to do 「呼び捨て」(yobisute), meaning to "throw out how-to-call-someone," and should only be done through consent by both parties and indicates a high level of closeness or conveys a lack of respect (for example, the times when Rei calls Usagi without a suffix, or when Usagi calls Chibi-usa without a suffix).
Characters usually use yobikata for each other, most of the cases are the common "-chan," (Usagi-chan, Ami-chan, Rei-chan, Mako-chan, Minako-chan, Mamo-chan, Chibi-usa-chan, etc.) and are sometimes used for the sailor senshi forms (V-chan, Venus-chan, etc.). "-P" is a rather uncommon suffix in Japanese culture; I have lived in Japan for years and never heard someone use it (though, admittedly, I don't spend time at Japanese high schools). In contrast, "-chan" is everywhere in society.
It should be noted that this "-P" suffix is a wholly different "-P" suffix than the "-P" suffix used to abbreviate the word 「プロデューサー」(purodyuusaa = producer) in arenas such as the vocaloid industry. In Bishoujo Senshi Sailor Moon, the civilian disguise form of Sailor Iron Mouse, Nezu Chuuko, who works for Ginga Terebi (Galaxy TV), is called "Nezu-purodyuusaa," never called "Nezu-P."
As you note, the only two within the series to receive the "-P" suffix are Minako and Luna-P. One reason for each case overlap, but there are two points of difference.
When Sailor V(enus) first appears, she is a mysterious, unknown, admired enigma to Usagi and the other others. However, over time, they get to know Minako. While Usagi loves all of her friends deeply, she comes to feel a particularly special bond with Minako because she views Minako as more like herself than the others. (In the early manga and anime, in what ways Minako is more similar to Usagi than Makoto is to Usagi or than Rei is to Usagi is not so clear but, as the series goes along, they are portrayed as a kind of kindred combo. This kindred affinity is portrayed in the anime as well [for example, chasing after Haruka-san or lamenting failing a class], but in the anime the Usagi-and-Rei relationship is portrayed more centrally.) Because Usagi feels a more special bond with Minako, she calls her "Mina-P," a suffix she doesn't use for anyone else.
As this Sailor Moon blogger explains,
Translated: "What I love the most is how Usagi calls Minako 'Mina-P.' You can feel how close of friends they are. What they call each other is an important point that shows human relationships and their personalities. Though they change it up once in a while. (ﾟAﾟ;)" [translation mine]
In contrast, the suffix serves three purposes for Luna-P, Chibi-usa's cat face ball device.
One reason is the same: to demonstrate that Chibi-usa considers this device her close friend.
Secondly, the "-P" suffix in Luna-P's name is onomatopoeia, (in Japanese, 「擬音語」[giongo]). 「ピ」(pi) or, usually in repetition, such as「ピピピピピ」(pipipipipi), is a Japanese sound effect that means "beep beep beep." This sound effect often appears in the manga, and Luna-P itself only ever communicates in beeps rather than words (excluding transmitting voice chat between Chibi-usa and Puu). Luna-P is a beeping device, so the sound of the pronounced "-P" suffix here additionally conveys the electronic nature of the item.
Thirdly, other characters adopt calling this device "Luna-P" even though they do not feel that personal affection toward it, most likely because to speak of it with yobisute would simply be to say the name "Luna" - the name of a different prominent character who is always referred to and addressed with yobisute - and thereby possibly confusing.