The PostgreSQL dev team modified their release naming after 9.6, making 10.x the next major release, then 11.x, etc (currently at 15, having just released 15.2, 14.7, 13.10, 12.14 and 11.19).
Esri generally supports the "current" build (as of development start), plus three back. So ArcGIS 10.6 (the oldest currently maintained release) supports PG 10, PG 9.6, PG 9.5, and PG 9.4. Each supported release has a "minimum" build (the one developed against), but minor updates are also supported (unless explicitly noted due to breaking change). For ArcGIS 10.6.1, those minimums are PG 10.3, 9.6.8, 9.5.12, and 9.4.8 (all of which are no longer supported by the PG dev team). PG 11 is not supported (at all) with ArcGIS 10.6.x.
Esri support for PostgreSQL generally increases to the "most recent" release with each ArcGIS major release to the active PostgreSQL release, so ArcGIS 10.7+PG11, ArcGIS 10.8+PG12, and ArcGIS 10.9+PG13. ArcGIS 11.0 still only lists PG13, but PG14 might be added with the terminal 11.0 snapshot (and/or 11.1). PG 15 is not yet shown as supported.
Note that there was a breaking change between PG 11 and PG12, where support for WITH OIDS=TRUE was removed. The ArcGIS releases which support PG12 and higher are documented as needing a pg_dump|pg_restore migration from PG11.
If you're currently at PG 10.3 with ArcGIS 10.6.1, you probably need to upgrade ArcGIS to 10.7.1 or 10.8.1 (to make PG11 and/or PG12 available), then upgrade PG10 to PG11 (or PG 12), then upgrade ArcGIS again to 10.9/11, before upgrading PostgreSQL to 13.x. I wouldn't recommend changing both ArcGIS and PG releases at the same time without a full export/import migration (which is useful at times, but requires a lot of work).
- V