Considering how much higher they are than every other country, was there some record keeping shenanigans going on?
The high-scoring German aces flew many operational sorties in a target-rich environment. Most of their accepted claims are reliably confirmed. An unknown, but fairly small, fraction of their claims are unconfirmed and therefore less reliable.
We can pick a high-scoring German pilot whose operational record is reliably-known to compare with some other pilots: Johannes Steinhoff, with 176 victories (152 on the Eastern Front). To achieve these, he flew 993 operational sorties, for an average of 1 victory for every 5.6 sorties. He was shot down 12 times.
Another high-scoring pilot who also flew against the Soviets is the Finnish pilot Ilmari Juutilainen, with 94 confirmed victories (and 32 claimed but unconfirmed) in 437 sorties. This is one confirmed victory per 4.6 sorties - faster than Steinhoff's victories. Another Finnish pilot, Kyösti Karhila, gained 32.25 victories in 304 sorties, one victory per 9.4 sorties.
Opposing these pilots were Soviet pilots such as Lev Shestakov, with 26 victories on the Eastern Front from over 450 sorties; only about 100 of those sorties involved meeting enemy aircraft. Taking into account his reduced opportunity due to fewer targets, his scoring rate is comparable to Steinhoff and the Finnish pilots above. Alexander Pokryshkin gained 53 confirmed kills from 550 sorties, quickly-scoring if he also only encountered enemy aircraft on approximately 1/4 of those sorties.
Among British pilots, Neville Duke gained 27 victories from 486 sorties. Johnnie Johnson gained 38 victories from about 700 sorties, in 57 of which he encountered enemy aircraft.
Allied pilots could score victories quickly, if they met the enemy - Johnnie Johnson above is one example. Another example is US Navy pilot David McCampbell, who flew in the very target-rich environment of the "Marianas Turkey Shoot" and the Battle of Leyte Gulf. Both battles involved flying against poorly-trained enemy pilot who flew inferior aircraft. On two sorties on 19th Hune 1944, in the "Marianas Turkey Shoot", he shot down 5 and 2 Japanese planes, and in the Battle of Leyte Gulf, he shot down 9 Japanese planes in a single sortie (he and his wingman attacks a Japanese force of 60 aircraft, together shooting down 15 of them).
Considering their large number of sorties, their plentiful targets, and that many of their victories were on the Eastern Front against relatively unskilled pilots and/or inferior aircraft, the German high victory scores are realistic. If they're inflated, they're not greatly inflated.
Overall, German fighter pilots claimed to shoot down 45,000 aircraft on the Eastern Front, and 25,000 British and US aircraft. The Soviet air force lost about 46,000 aircraft in combat, and the UK and US together lost about 80,000 aircraft in combat.
WWII German media claimed much higher numbers of enemy aircraft shot down than the number of confirmed victories credited to pilot. Individual pilot victory totals shouldn't be judged on the basis of inflated claims in the media.