Can anything reveal that 2 independant entries were created or updated at the same time?
Rows inserted at the same time, particularly in the insert-only use case such as yours, will very likely reside next to each other on disk and as a result will have tuple IDs indicating that. They will remain so until you CLUSTER the table using  your unique_random1 column as the index key. You will need to run CLUSTER periodically to shuffle newly inserted rows.

Another bit of information that potentially correlates inserted rows is their transaction ID (XID), which might remain unchanged after the original commit, unless these rows at some later time are updated by separate transactions. I don't believe CLUSTER changes row XIDs.

Basically the same information -- the transaction ID, along with the row data, -- will also be stored in the write-ahead log (WAL). WAL segments will get overwritten at some point, depending on your database's checkpoint frequency.

TL;DR: You probably shouldn't rely on Postgres internals for reliable deidentification of your data. 



1 -- Although, what's truly random, cannot have guaranteed uniqueness. You probably meant some sort of a hash.