To copy into an existing table, use the geoprocessing Append tool, NOT copy features. Append has its own Field Map control in its dialog, separate from Transfer Attributes in the project's Editor tab.
Clicking Edit on the field map brings up Field Properties where most of the action is.
The input to Append can be a layer in your map with a selected set of records. Those selected records are what are appended into the target dataset.
It seems like your mixing the Edit/Table ribbons in the project with running the geoprocessing tool. I don't use those ribbon tools much -- I think I tried the transfer attributes tool once and got confused 🤔 so I went back to using Append.
Append only works with similar geometry. You can append point features into another point feature class, but not lines/polys. You can append stand-alone table records into another stand-alone table. Are you trying to append a record from a stand-alone table (a table w/o a shape column) to a feature class?