While i haven't seen the anime yet i am confident to say that the anime isn't an adaptation of any if the game's plot rather it makes references to the games
like the with the 2 Noire's one is the alternate one from the Ultradimension
as for the order of the games not all of them relate to one another as most are in different dimensions however Mk.II/Re;Birth 2 are Victory/Re;Birth 3 are linked together as the Wikia Says
A few years have passed since the Deity of Sin's destruction.
as for the different Dimensions
1st Hyperdimension - Hyperdimension Neptunia/Re;Birth
in this Arfoire was a single goddess before she and Histoire made Neptune, Noire, Vert and Blanc. also Histoire has near unlimited power because of her role as a Tome
also in the true ending of the original PS3 game (have yet to confirm with Re;Birth) the CPUs give up their divinity to eradicate all of the monsters across the world and became regular humans
In Re;Birth when characters like MAGES., Tekken and Falcom talk about being from another world i suspect this is the Ultradimension/2nd Hyperdimension as i first learned of these characters from the announcements of other human characters in Victory before it's release
2nd Hyperdimension - Hyperdimension Mk. II/Re;Birth 2: Sisters' Generation, Hyperdimension Neptunia Victory/Re;Birth 3: V Generation, Megamdimenstion Neptunia VII
Histoire isn't as powerful as he is in the first instead being one of 4 Oracles. in Mk.II IF seemed to have a crush for Nepgear rather than Vert. in the True Route the party goes to where one of the former Goddesses of Planeptune are buried who they speak to about DOS.Arfoire.
Also aside from the first game/Re;Birth i believe all games which refer to the Hyperdimension are set here as according to the Wikia page of the Hyperdimenstion
The Hyper Dimension is a dimension of the Hyperdimension Neptunia series and appears in every title of the main series since Hyperdimension Neptunia mk2.
also while i haven't found where VII links with with Mk. II/Re;Birth 2 and V/Re;Birth 3 it's probably safe to assume that VII would get set after V/Re;Birth 3 as VII isn't pronounced as 7 (roman-numerals for VII) but as V2, short for Victory 2
Megadimension Neptunia VII (新次元ゲイム ネプテューヌVII Shin Jigen Game Neptune Victory II, "New Dimension Game Neptune Victory II" in Japan) is a Japanese role-playing game in the Hyperdimension Neptunia series of role-playing games, developed exclusively for the PlayStation 4.
Source: Wikipedia - Megadimension Neptunia VII
Ultradimenstion - Hyperdimension Neptunia Victory/Re;Birth 3: V Generation
Alternate dimension that Neptune is swept away to which represents a late 80s version of the 2nd Hyperdimension Gamindustri. more can be read up on the Wikia Page
Alternate Ultradimension - Hyperdimension Neptunia: Producing Perfection
Hyperdimension Neptunia: Producing Perfection takes place in alternate Ultradimension that is the same as the Ultradimension of Hyperdimension Neptunia Victory in terms of geographical structure. In this dimension, residents who typically are not from this world such as the Four Felons exist as well as numerous others.
Source: Hyperdimension Neptunia: Producing Perfection - Setting
the 4 Felons it's referring to are ASIC's Criminals of the Free World, CFW.Magic, CFW.Judge, CFW.Trick and CFW.Brave
Gamarket - Hyperdevotion Noire: Goddess Black Heart
Unlike previous titles of the Hyperdimension Neptunia series. Chou Megami Shinkou Noire Gekishin Black Heart takes place in a different world known as "Gamarket." In this world the nations of Planeptune, Lastation, Lowee, and Leanbox still exist with their respective goddesses and the geographic structure is mostly the same as the Hyperdimension and Ultradimension but still visually different at the same time.
Source Hyperdevotion Noire: Goddess Black Heart - Setting
Alternate 1st Hyperdimenstion from Re;Birth? - Hyperdimension Neptunia U: Action Unleashed
Hyperdimension Neptunia U takes place in a Gamindustri much like the the one of Hyperdimension Neptunia Re;Birth1. The geography is entirely the same and is comprised of floating continents; each having one of the four nations: Planeptune, Lastation, Leanbox, and Lowee.
Source: Hyperdimension Neptunia U: Action Unleashed - Setting
while the Wikia says "much like" this could be because it may be missing Celestia. i have yet to play it to determine this
MegaTagmension? (Currently unknown) - MegaTagmension Blanc + Neptune VS Zombies
The game's setting is very different to the previous games. In this game, set in another alternate universe, the girl are not CPU/Goddess but instead high school students. They try making a zombie movie when they accidentally summon zombies to their world.
Source: MegaTagmension Blanc + Neptune VS Zombies - Setting
Zero and Heart Dimensions - Megamdimenstion Neptunia VII
The Zero Dimenstion is an alternate dimension nearing destruction where Neptune and Nepgear end up first.
As for the Heart Dimension it is,
An amalgamation of both Zero Dimension and Hyper Dimension. In order to track down the mastermind causing the troubles of both worlds, the CPUs travel to a strange, new dimension.
Soruce: Megadimension Neptunia VII - Story - Heart Dimension Neptunia H Trilogy Finale: Into Legend
As for what Compa says in the Lily Rank Event it would be true on 2 levels as
- the 2nd Hyperdimension for Re;Birth 2 is different from the 1st one in Re;Birth thus it would have a different history
- in the Ultradimension IF and Compa appear in this too but as babies (so one can consider the Ultradimension as an alternate past to the 2nd Hyperdimension)
As for Recommended Play/Watch Order:
Hyperdimension Neptunia (PS3)/Re;Birth 1 - this introduces you to the main reoccurring characters such as the CPUs, Compa and IF and Arfoire
Given that Mk. II/Re;Birth 2, V/Re;Birth 3 and VII are initially linked i would play them in that order after Re;Birth
Producing Perfection, Action Unleashed and Goddess Black Heart you can probably play at any time after Mk. II/Re;Birth 2 because to my knowledge there are no characters from V/Re;Birth 3 in there
Blanc + Neptune i would play after V/Re;Birth 3 because Plutia and Peashy appear in the List of characters and more than likely they will be like that from the Ultradimension
i would watch the anime after V/Re;Birth 3 for the same reason as above because of Plutia and Peashy who apparently appear in the anime (i have yet to watch it)
NOTE: all of this does not take into account DLC characters as Re;Birth 1 features Plutia and Peashy but they are not involved with the plot (rather it might be confusing as to why Plutia may seem clam and kind of a ditz but Iris Heart is a Dominatrix)
NOTE 2: This does not take into account any of the Manga because i am unfamiliar with them at this point in time