From your use case, I understood that you want to search for a particular metadata attribute, like a Tag field, PII, right?
For tagged assets
If you don't care about the template name. You could use the tag:x search facet.
So if all your templates, data_gov_template, data_curator_template, data_etl_template, all contain the same Tag field name, has_pii, you can search using:
tag:has_pii and this will return all assets with that metadata attribute, no matter what the template name is.
For columns
You can use the column:x search facet to match a substring of the column name in the schema of the data asset. Which does not support nested columns yet.
For labels
You can use the labels:bar search facet for data assets that have a label (with some value) and the label key has bar as a substring.
You are also able to search on their values. So yes, the metadata/attributes and values are searchable.
But it is not a regex kind, it is a substring match when the search facet uses colon :, like labels:bar or an exact match when the search facet uses equals =, like type=table.