What is the official classification and naming system of objects in the universe(, or what would my URI look like)? Poetic human centric approach:

Universe/Laniakea/Virgo/MilkyWay/Sol/Earth/Humans/AndreasT


Little bit more technical:

Universe/0/SuperCluster/0/GalaxyCluster/0/Galaxy/0/StarCluster/0/Star/0/Planet/3/../HomoSapiensSapiens/2834929849289


More ... multiversal?

Or is there something in the spirit of the rebasing of SI units to universal concepts that makes us not be the origin(hah!) of the system?
Objects aren't named like that.

Either objects are given "names" that have no structure. For example, an asteroid may be named "Vesta".

Or they are given essentially meaningless catalogue numbers. For example, a star might be catalogued as HD 138987. It is number 138987 in the Henry Draper catalogue. Some major stars do have their constellation as part of the name. Alpha Centauri is the brightest star in the Centaurus constellation.

Comets are often just named according to the year of discovery 

Other objects are named by their position in the sky, using polar coordinates.

There is no URI-like system in use.

The reason for this is that there is no natural hierarchy. When naming species of animal, they naturally group into genus, family, order, class as a result of evolution. No such structure is found among astronomical objects, so a hierarchical approach is not appropriate.