No, you cannot update the values in a query layer, nor add writable fields to a query layer. (In general... technically this is not always true, but for my understanding of this case, it is true.) The documentation for ArcGIS Query layers states, "Query layers are read only when accessed directly from a database connection."
The main difference between the 'Multi' and non-Multi 'Extract' to Points tools is that the (single) 'Extract Values to Points' tool writes its output to a new output feature class, but the 'Extract Multi Values to Points' tool updates the input feature layer (adding field(s) to it and writing the values to those field(s)). It even displays the warning "This too modifies the input data" at the top of the tool pane.
Query layers cannot be modified, therefore they cannot work with the 'Multi' tool. You could use the underlying feature class (that the query layer uses as its spatial component) as the input, rather than the query layer itself.
Query layers are simply a read-only (spatial) database view. They are not like a feature layer, which has a simple feature class as its data source.
Note that "proper write permissions to the database" are not the real issue here. The issue is that query layers are not writable. You could use a feature layer (feature class) which is stored in the same database, and then, so long as you had write permission to that feature class, it should work.