First make sure your Buildings have a unique id (which does not contain a | character, you will see why below). You can add it for example via field calculator by adding a new field with the expression $id if they do not have one already.

Then run "Polygon Self-Intersection" from SAGA Processing tools and choose this ID as Identifier:

Which will create a new layer, where the overlapping Parts become their own polygons.

This new layer will contain a new ID field, containing the IDs of their original buildings splitted by a | sign.

Now open the field calculator of this new layer and calulate a new field with this expression:

if(regexp_match("ID",'\\|') > 0,regexp_substr("ID",'[^|]*'),"ID") to get the second ID as mergeID or if(regexp_match("ID",'\\|') > 0,regexp_substr("ID",'(?<=\\|).*'),"ID") to get the first ID as mergeID:

This expression looks up in your "ID" field, wheter it contains a | character. If so, it chooses the ID before or after the | character. If it does not contain this character, it will just copy over the ID to the mergeID.

Then run "Dissolve" from processing toolbox and choose this mergeID as dissolve field:

Your final result:

First make sure your Buildings have a unique "id" (which does not contain a | character, you will see why below). You can add it for example via field calculator by adding a new field with the expression $id if they do not have one already.

Then run the "Polygon Self-Intersection" from SAGA Processing tools and choose this "id" as Identifier:

Which will create a new layer, where the overlapping Parts become their own polygons.

This new layer will contain a new "id" field, containing the "id"s of their original buildings split by a | sign.

Now open the field calculator of this new layer and calculate a new field with this expression:

if(regexp_match("ID",'\\|') > 0, regexp_substr("ID",'[^|]*'), "ID")

to get the second "ID" as "mergeID" or

if(regexp_match("ID",'\\|') > 0, regexp_substr("ID",'(?<=\\|).*'), "ID")

to get the first "ID" as "mergeID":

This expression looks up in your "ID" field, whether it contains a | character. If so, it chooses the ID before or after the | character. If it does not contain this character, it will just copy over the ID to the "mergeID".

Then run the "Dissolve" from processing toolbox and choose this "mergeID" as dissolve field:

Your final result: