How to merge polygons that touch and share some features
--
You only need [`ST_ClusterIntersectingWin()`][1].  
If you're below PostGIS 3.4.0 but above 2.3.0, you can use [`ST_ClusterDBSCAN()`][2] with target distance of `0`. [Here's a related thread on SO from recently][3].
```pgsql
create table hrl_tst.your_shapes_back_together as
with both_tiles as (table hrl_tst.tile1 union all table hrl_tst.tile2)
,clustered as (
    select *,ST_ClusterDBSCAN(geom,0,1)over(partition by cat) c_num
    from both_tiles)
select cat
      ,array_agg(unid order by unid)--dissolve same `cat`s ignoring `unid` diff
      ,st_makevalid(st_union(geom)) as geom --it's healthy to `st_makevalid()`
from clustered
group by cat,c_num;
```
I spawned around 190 polygons assigned to 3 `cat` classes (it's zoomed in on ~120):

[![enter image description here][4]][4]

And after running the query above, all polygons that share the `cat` feature and `ST_Touch()` (actually, they can `ST_Intersect()` in any way) directly or indirectly (through a neighbour they touch, or a chain of touching neighbours' neighbours) get merged.

[![enter image description here][5]][5]

It can stitch shapes broken along a tile boundary back together, but as you can see it works with arbitrarily shaped splits just as well - I just smashed my two squares into pieces, threw some away and shook everything hard to make mess\*.

This does not attempt to simply `ST_Union()` all Polygons that share the same `cat` feature - that would be trivial, but result in one large MultiPolygon for each `cat`. The disjoint brown, blue and green shapes above are separate, standalone shapes unless they touch somehow.

If your `unid` field is unique but spans the sets, meaning that the same shape's split between the two tables but holds the same unid in both, and you don't want to merge shapes of differend `unid`, you can add `unid` anywhere on the `group by` list and swap out `array_agg()` for just `unid`.

***
\*<sub>(Make some squares, spawn 90 random points in each, subdivide them with Voronoi cells around those, discard random 30%, wiggle the rest furiously with `ST_Translate()` with `random()` x&y offsets, then trim them with `ST_Difference()` so that they can touch, but not overlap).</sub>


  [1]: https://postgis.net/docs/ST_ClusterIntersecting.html
  [2]: https://postgis.net/docs/ST_ClusterDBSCAN.html
  [3]: https://stackoverflow.com/a/78228340/5298879
  [4]: https://i.sstatic.net/nHMUpVPN.png
  [5]: https://i.sstatic.net/lQiMkUy9.png