LMDB is a key-value database. The Euclides DB build on top of a existing key-value database, LevelDB (could be LMDB) as the persistent layer.
Im not familiar with EuclidesDB, but from what i've understand from the documented architecture it uses gRPC to expose a RPC interface over the internet, and abstract away the objects in the domain of machine learning (like images) tied to PyTorch.
Anyway, its a higher level concept and architecture modeled on top of a key-value storage like LMDB, which is a lower level building block for this product. LMDB being comparable to LevelDB, not EuclidesDB.
(It looks that the author is here, so please correct me if im mistaken in any way)