I am aware of popular initiatives such as the Wikipedia articles listing astronomical objects on the basis of some criteria but... do some kind of systematic indexing, or catalogue, of objects found, potentially via machine learning, in each and every astronomy images, exist ? So as to facilitate the search for this object within other pictures of it ? Then for example, when an astronomer look a some faint object within the latest jwst deep field, it can immediately look up if that object is already listed in the Hubble deep field of the same region of space ?
You can use the SIMBAD Astronomical Database, and query the cluster and its members. Unfortunately, the cluster in JWST's Deep Field (SMACS 0723) was not yet covered by any major survey and its member galaxies have no data yet in SIMBAD as of the moment.
However, you can query for all known objects around it. Here is a list of objects around 10 arcminutes of the cluster:
Alternatively, and even better, you can use the NASA/IPAC Extragalactic Database, though this is much harder to use. Here is a list of all objects within 1 arcminute of the cluster:
Keep in mind though that this lists all infrared sources. They are not yet classified (whether they are galaxies, or stars). Only the WISE spacecraft has scanned the area in detail so far.
If you wanted to cross-check sources found in the JWST image with those found in the HST deep images, then the HST data and catalogue can be found in the "RELICS" database.
More generally, imaging surveys that cover a large part of the sky will usually publish their results as a catalogue of sources. These are identified by a name (usually an IAU standard designation) and a position. All these catalogues are available online at somewhere like the CDS in Strasbourg - they even offer an online query tool, called Vizier that will search these catalogues around user-defined positions (NB There are thousands of catalogue entries within 2 arcminutes$^\dagger$ of the nominal position of SMACS J0723 - there are infrared sources from the Vista Hemisphere Survey, 2MASS, DENIS and WISE; far infrared sources from Herschel and Planck; optical counterparts from Gaia, TESS and the various Guide Star Catalogues; X-ray sources from Swift and XMM-Newton; and others; it also to my surprise includes the RELICS HST survey catalogue I mentioned above!).
The catalogue entries within any region can be saved in various machine-readable formats (fits, votable etc.). It is up to the end-user to decide whether a faint object in their image corresponds to an object in one of these catalogues.
$\dagger$ i.e. if you just search in a 2 arcminute radius around 07 23 19.5 -73 27 15.6
NB 1: Wikipedia is not a source of data.
NB 2: The Hubble Deep Field(s) cover a tiny fraction of the sky.