Sibling Triangle Pairs

For a = 14921875 > b = 14687500, we have exactly 50 sibling pairs with common side lengths a and b, namely:
(a, b, 804375) and (a, b, 29599375) : area = sqrt(32419309541000976562500000).
(a, b, 1842135) and (a, b, 29552945) : area = sqrt(182225637700074127453140000).
(a, b, 2011875) and (a, b, 29541875) : area = sqrt(217769218524695434570312500).
(a, b, 2062625) and (a, b, 29538375) : area = sqrt(228993128773469156250000000).
(a, b, 2479275) and (a, b, 29506325) : area = sqrt(331461957498265003626562500).
(a, b, 3046875) and (a, b, 29453125) : area = sqrt(500319153070449829101562500).
(a, b, 3097471) and (a, b, 29447847) : area = sqrt(516987963491157490115899584).
(a, b, 4078125) and (a, b, 29328125) : area = sqrt(891056246995925903320312500).
(a, b, 4296875) and (a, b, 29296875) : area = sqrt(987429767847061157226562500).
(a, b, 5271825) and (a, b, 29137225) : area = sqrt(1471671982901439733837500000).
(a, b, 5321875) and (a, b, 29128125) : area = sqrt(1498867208641433715820312500).
(a, b, 5881667) and (a, b, 29020269) : area = sqrt(1817880201964237609922206644).
(a, b, 6290625) and (a, b, 28934375) : area = sqrt(2067588189459228515625000000).
(a, b, 6340325) and (a, b, 28923525) : area = sqrt(2098858330238817432501562500).
(a, b, 6506375) and (a, b, 28886625) : area = sqrt(2204749758016921500000000000).
(a, b, 7515625) and (a, b, 28640625) : area = sqrt(2892834314346313476562500000).
(a, b, 8065625) and (a, b, 28490625) : area = sqrt(3297340218328857421875000000).
(a, b, 8466891) and (a, b, 28373963) : area = sqrt(3604171173981076126656478404).
(a, b, 8515625) and (a, b, 28359375) : area = sqrt(3642064332962036132812500000).
(a, b, 9060025) and (a, b, 28190175) : area = sqrt(4073930349838387765650000000).
(a, b, 9666085) and (a, b, 27988155) : area = sqrt(4571344112209274638609402500).
(a, b, 10203125) and (a, b, 27796875) : area = sqrt(5024331455469131469726562500).
(a, b, 10250865) and (a, b, 27779305) : area = sqrt(5065073958734665315195140000).
(a, b, 10641875) and (a, b, 27631875) : area = sqrt(5401259997789050903320312500).
(a, b, 11171875) and (a, b, 27421875) : area = sqrt(5862779170274734497070312500).
(a, b, 12126875) and (a, b, 27013125) : area = sqrt(6703977212172559204101562500).
(a, b, 12281855) and (a, b, 26943015) : area = sqrt(6840837711095270483001240000).
(a, b, 12328125) and (a, b, 26921875) : area = sqrt(6881684680223464965820312500).
(a, b, 13219375) and (a, b, 26495625) : area = sqrt(7664426740542612304687500000).
(a, b, 13264875) and (a, b, 26472875) : area = sqrt(7704046824662650157226562500).
(a, b, 14140625) and (a, b, 26015625) : area = sqrt(8455342054367065429687500000).
(a, b, 14185299) and (a, b, 25991293) : area = sqrt(8492956242325263688637021364).
(a, b, 14334375) and (a, b, 25909375) : area = sqrt(8617866359930419921875000000).
(a, b, 15234375) and (a, b, 25390625) : area = sqrt(9348378181457519531250000000).
(a, b, 15720375) and (a, b, 25092625) : area = sqrt(9722178209954832398437500000).
(a, b, 16072925) and (a, b, 24868275) : area = sqrt(9982291390563929549470312500).
(a, b, 16115625) and (a, b, 24840625) : area = sqrt(10013106389089965820312500000).
(a, b, 16258059) and (a, b, 24747637) : area = sqrt(10114761888674890613164587204).
(a, b, 16590873) and (a, b, 24525761) : area = sqrt(10345158227887036747470978624).
(a, b, 17116125) and (a, b, 24162125) : area = sqrt(10686583034378413497070312500).
(a, b, 17578125) and (a, b, 23828125) : area = sqrt(10961894541978836059570312500).
(a, b, 17953125) and (a, b, 23546875) : area = sqrt(11166308746576309204101562500).
(a, b, 18403125) and (a, b, 23196875) : area = sqrt(11386948330566787719726562500).
(a, b, 19205475) and (a, b, 22537075) : area = sqrt(11706136084462849641970312500).
(a, b, 19334625) and (a, b, 22426375) : area = sqrt(11747850889345133062500000000).
(a, b, 19373125) and (a, b, 22393125) : area = sqrt(11759737885627637329101562500).
(a, b, 19687215) and (a, b, 22117495) : area = sqrt(11847047633475238763118540000).
(a, b, 20109375) and (a, b, 21734375) : area = sqrt(11936086887359619140625000000).
(a, b, 20146685) and (a, b, 21699795) : area = sqrt(11942328831927507139301752500).
(a, b, 20859375) and (a, b, 21015625) : area = sqrt(12007659673690795898437500000).

The explanation is that  a² + b² = 438385009765625 =: N  is the smallest number that is the sum of two positive squares in exactly 51 ways, cf. OEIS.org/A016032.
My PARI/GP program  sum2sqr(N)  given in OEIS.org/A133388 allows to get all these decompositions, and in particular, the given (a,b) = (14921875, 14687500) is the one with the largest smaller part, or otherwise said, with the least difference |a-b|.
 
Then,  2 N  is also the sum of two positive squares in exactly 51 ways, as one can check with the same sum2sqr program. On the other hand,  2 (a² + b²) = c² + d²  (*)  is the condition that the areas of the triangles  (a,b,c) and (a,b,d)  are equal.
Indeed, the area of a triangle (a,b,c) is given by AREA(a,b,c) = sqrt(a²b²/4 - (a²+b²-c²)²/16), and  AREA(a,b,c) = AREA(a,b,d)  leads to the above equation (*).
[We must have (a²+b²-c²)² = (a²+b²-d²)², and for distinct  c, d  the two expressions inside the parentheses cannot be equal, so they must be the opposite of each other, viz: a²+b² - c² = d² - (a² + b²), assuming WLOG  c < d.]

Now,  2N  has 51 decompositions as sum of two squares c² + d², but the decomposition with the smallest part c,  2 N = 234375^2 + 29609375^2,  would correspond to degenerate triangles (a,b,c) and (a,b,d) with zero area (since c = a - b and d = a+b), and we must exclude these. (This is the same case as the example a=2, b=3 you give for the case where there are no sibling pair at all -- since 2 (2² + 3²) = 26 = 1² + 5² corresponds to c = b-a and d = a+b.)
So there remain exactly 50 decompositions which correspond to sibling pairs with common side lengths a and b.