Dissolving polygons by specific field, but only neighbors polygons

If the polygons you want to dissolve are few, then you can merge them manually using Merge tool, as you can see below:

Start editing → Select the polygon you want to merge → it will prompt you a table to choose which field you want to use. Then select that feature and click OK.

ArcGIS has the same tool. Go to Editor → Start editing → Select the polygons you want to merge → Go to editor and select Merge.

Right-click the toolbar panel and Select Advanced Digitizing toolbar, as you can see below:

You could try to run Multiparts to Singleparts function after dissolving. Features that do not touch should become singleparts.