Main reason is that other militaries had field guns/anti tank guns in this class that Germans did not.

88mm flak in anti tank role was improvisation and using it in role it wasn't intended for. 88mm flak was by design anti aircraft gun, it was effective anti tank weapon because round had great initial energy, which was required to propel it to desired height. As such it was also good AT weapon. Early in the war Germans simply lacked anti tank weapons that could be used against existing tanks. In 1939 and 1940 main German AT gun was 37mm Pak 36 which was hopelessly inadequate for even medium tanks. The replacement was 50mm Pak 38 which didn't enter service until early 1941 and was able to penetrate T-34 armour at close range. Which was still inadequate and gun was in turn superseded by 75mm Pak 40 that started to enter service in early 1942 (some early production examples were rushed to the front in late 1941) which then became standard German AT gun until the end of the war as it was capable of defeating Allied tanks until late war heavy ones. So for Germans using what was designed as AA gun in AT was simply necessary because there was nothing else that could do the job.

And as much as 88 is mythologised and gained fame it was not in ideal AT weapon. It was big, it was bulky, it was heavy, it required significant crew and it took time to be set up or relocated. All of which made it less than ideal for the type of mobile warfare German doctrine called for. It was used extensively because it was one of few weapons that could do the job and one that was available from start of the war.

On the other hand Red Army had plenty of 76mm field guns, the Zis-3 and F-22, which could easily be used in AT role though F-22 was designed to be field gun/AA gun. As field guns they were designed to be easily moved and set up, making them much better choice for this role than 88s. Captured 76mm guns, both Zis-3 and F-22s were used on German Marder series of Panzerjäger. Red Army did have AA gun similar to 88, the M1939 (52-K) but wasn't used in AT role comparable to 88s because there was no need as 76mm did the job. Gun was later modified for T-34-85 tank and Su-85 tank destroyer, though, same as 88s were.

Similarly Western Allies did have existing AT guns, of course upgraded and replaced by newer models as war progressed, so no need to press existing AA guns into AA role. UK in early war had different issues with lack of guns and so 3,7inch (94mm) was considered for AT use in North Africa but not widely because issues I mention 88s had were even worse with this gun. As UK rather used dedicated AT guns (6 and later 17 pounders) who being designed for the role didn't have issues plaguing AA guns used in AA role.

So to answer your question, it wasn't copied because there was no need to copy it as Allies had better weapons for the AT role 88s were used in.