sorry for leaving your good comment and question without reply for some time - I feel that the LCA community does not have a clear answer here yet.
Of course, in some restricted, more precisely defined application contexts, it happens that one specific background database is mandatory and only possible to be used (Environmental Footprint, e.g.) which helps to align results of different studies done, but on the other side means that any process and product not available in this "prescribed" database cannot be addressed (if you do not model it yourself) and thus has zero impacts, which is also not perfect if these products and processes are not clearly negligible for the study (which, without knowing them, is always hard to decide).
For Environmental Product Declarations, there have been also specific databases proposed, with often funny and annoying outcomes (GaBi databases prescribed for EPDs in Germany, ecoinvent in some other countries). This has been softened recently though, so that you can use ecoinvent e.g. also in German EPDs (I am shortcutting a bit the details).
In general, you can consider if you accept databases that only distribute aggregated system processes, where you in the end e.g. need to trust what is done by a German SME (that was recently purchased by a US company), without possibility to really check this work, and where you cannot change anything in the supply chains yourself, or if you don't accept only-system-process databases. Then, in UN GLAD, a data quality concept has been developed which helps to find and assess datasets, and this could in principle be applied for entire databases too but is not available yet. However, in principle, these data quality indicators assess whether the data fits to your modeling approach and to your product and process that you need, and this is of course already now applicable for the database selection.
But overall, as said in the beginning, I am afraid not a too clear answer probably..