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Eotain and Shurdlu may (or may not) be indistinguishable, but they are not in any case identical; they have separate memories and reactions. One of them is amused that they at first mistook Agatha for a mere actor, the other is dismayed. One finds Dingbot Prime delightful, the other is disgusted. Whether it is the same one with the more carefree attitude in each case and which has which name has not been stated in canon.
In a certain Othar twitter, Eotain states Shurdlu was "the more pious one", able to "talk her through her moments of doubt". Two tweets before, Othar states that Eotain is "a person who likes to talk". And later on, in a battle in a restaurant, Eotain says she had a fine time, and really liked the waffles. Based on that, we can tentatively speculate that the one who always laughs at things would likely be Eotain and the one to take their mission maybe a tad too seriously would be Shurdlu.
Relationship betwen them
Eotain and Shurdlu clearly aren't just "two random Geisters that happen to appear often together and crack jokes at their hability to spot "actors"✣ ": for starters, Eotain gets extremely upset ✣ , shouting for revenge ✣ when Shurdlu is killed (something she doesn't do when other Geisters fall in combat). Also, it's plainly more than just a "comrades in arms" relationship: in a very informative Othar twitter, Eotain defines Shurdlu as "her partner", says that Shurdlu would "talk [her] through her moments of [religious] doubt" , and even goes into deep grieving for her death, actually breaking into tears when Othar sings the Song of Memory for her. So, what's the exent of their "partnership"? There's nothing definite, but it's worth noting that when Agatha meets them for the second time, they were sharing a bunk.✣ Agreed, they were in a prison cell with likely a limited number of bunks, and there are other characters shown sharing their bed with no implication of a relationship, but still, when combined with Eotain's reaction to Shurdlu's death, it heavily implies that yes, they were in that kind of relationship.
Possibly relevant outside information
First see Wikipedia:ETAOIN SHRDLU, which already includes a (brief) note on the Girl Genius use.