I know there are different versions of R2 droids but are there multiples of each version of those droids? In other words were there any other R2-D2 droids manufactured in the galaxy or is there only one? For instance I can have the same model cellphone as someone else but my phone will have a serial number making it unique. Is this true for R2-D2 or is it the "D2" after the "R2" that makes the droid unique?
There were plenty of R2 type droids.
However, R2-D2 was a unique instance of that designation in existing canon. Googling for "other R2D2 units|droids" returns nothing; as an extra proof.
However, conceptually, there COULD have been other R2D2 units.
As I noted in my answer here, according to canon (Pablo Hidalgo - Internet Content Provider for starwars.com - Q&A in Star Wars Insider #58 (Page 97); as well as Disney canon "Rogue One novelization), the 4-letter designation is merely an owner extracting easy to use name from a much longer serial number.
From the various uses in canon, the "R2" part is normally the model number, and the "D2" portion is the serial number.
Other R2 units are seen on screen, as well as plenty of R5 units. R5-D4 is the unit that blows its motivator in Ep IV, for example.
Other R2 units from starwars.wikia.com's R2-series_astromech_droid and R2 Category page include R2-0, R2-4B, R2-5, R2-A3, R2-A5, R2-A6, R2-B1, R2-B3, R2-B4, R2-BDE, R2-C2, R2-C3, R2-C4, R2-C9, R2-D0, R2-D2, R2-D5, R2-D6, R2-D609, R2-D7, R2-D9, R2-E7, R2-F2, R2-F7, R2-J8, R2-K7, R2-KT, R2-L1, R2-L3, R2-M3, R2-M5, R2-MK, R2-N3, R2-P3, R2-PU, R2-Q2, R2-Q5, R2-Q5, R2-Q8, R2-QU, R2-R7, R2-R9, R2-RC, R2-RD, R2-S4, R2-T0, R2-T7, R2-V0, R2-V6, R2-W3, R2-X0, R2-X2, R2-X3, R2-X9, R2-Y9, R2-Z1, R2-Z13, R2-Z4. Of these named R2's, R2-D2, Both R2-Q5's and R2-KT are on film (KT is in clone wars, Q5 #1 in Ep V, #6 in Ep VI). A very few Extended Universe materials have R2 units with non-conforming names, such as B3-TA in The Clone Wars Adventures video game.
A Similar list of R5-units is seen as well; R5-D4 is seen in Ep IV, blowing his motivator... Several are seen on film, and they are quite popular in the Extended Universe materials. It's also noted that some R2 units are in R5 chassis, to cover up the gaff in Ep IV where Luke calls R5-D4 an R2 unit, saying "Hey! This R2 unit has a bad motivator!"
I believe that they are not unique and that there are many R2-D2s out there in the galaxy.
Lets assume that all R2 units have a R2-XX name where XX can be any number or letter. That gives us 67600 possible R2 units. Wookiepedia states that there are around 1.75 million planets in the Empire. Which means only 3% of them would have an R2 unit anywhere on the planet.
For such a common droid it seems very unlikely that they'd constrain themselves to selling them in such small numbers.
My suggestion would be that it works more like a software versioning system. (Major-Minor).
- R2 represents the model
- D2 represents the version of the droid
There is then some specific serial number identifying each droid (presumably much bigger).
As to why they're not referred to by this unique version number? Would you rather call your droid R2-D2 or R2-D2-57073805371805317857832075501?
Assuming that the names of R2 units follow the pattern R2-XX, where X is a character in A-Z or 0-9 (37 possible characters), there are 37^2 or 1369 possible R2-unit names, which certainly wouldn't be enough to make them unique across thousands of star systems. My best guess is that people tend to use the model number as a name as a short-hand, much as Luke referred to R5-D4 by color when referring to it as 'Red'.
In fact, R2-D2 is usually referred to simply as Artoo, which is a specifier used as a generic for all astromech droids.
The interesting thing is that C-3PO introduces R2 as R2-D2, without any further specificity, indicating that either C-3PO is using the human naming convention, or that R2-D2 is in fact unique.
It seems pretty clear that the answer is "yes", given the information others have dug up about the Star Wars universe. I wanted to chime in and suggest how this could be possible while still allowing many droids.
As evidenced by aramis' list above, some of the units have more than one digit (R2-Z13, R2-D609), and some have only letters (R2-MK) or digits (R2-5, R2-0).
To me, this implies that "D2" is something akin to a hex number (or rather, at least base 36, since we can see "Z" being used). "0xFF" is considered a number, not a letter designation, in hexidecimal; the missing numeral glyphs are replaced with letter glyphs. The second half of the designation is probably a numeric serial number in base 36, starting at R2-0.
That gives us serial numbers in sequence: R2-0, R2-1, [...] R2-9, R2-A, R2-B, R2-C, [...] R2-Z, R2-10, R2-11, and so on and so forth. If my math is right, that would make R2-D2 the 470th droid produced, and the highest droid number on @aramis' list (D609) is the 614,313th droid of that type. The millionth R2 unit would be R2-LFLS, still pronounceable by humans in a hurry. Bumping up to five digits would get you 60 million droids.
Considering that doesn't count alternate lines like the QT, R2-AG, or R2-R, (and considering that sci-fi writers have no sense of scale), I think that's more than enough serial numbers to be plausible.
There are dozens of R2-D2 units in the galaxy, otherwise 2-1B would not have his 'own' sw wiki page and 2-1B Medical Droid having its own.
If R2-D2 is a unique number then logic would say 2-1B is unique as well, but the medical droid in SW3 is apparently not the same one from SW5. If there are several 2-1Bs in the SW Universe, there's probably a few dozen R2-D2s running around.