It has to do with the way QGIS' spatial join is written. Based on the screenshots of it I see, there is no relationship parameter available. ArcGIS' spatial join does offer this capability.
You can't remove the function because it's a different... 'thing' or parameter of the join. The join has one attribute slot to fill, but multiple values to select from. Therefore it must use a function on the choices to get a single value or simply take the first one found. This is because all of your data has the same attribute names. For example the attribute in every source is called 'count'. If each of the yearly sources had the attribute named 'year-count' (where year varies between sources), then it would work.
Duplicating an identically named attribute is a different thing, based on the relationship of the join. The QGIS tool doesn't offer this from what I can see. The ArcGIS tool does using the Join Operation parameter (see linked help file above). If set correctly it would create a duplicate geometry for each instance of the attribute - meaning if you're creating points you get one point for each 'count' in all your sources, all stacked on top of one another.
I'm not clear on formats of your data and what your desired outcome is. It's possible that a spatial join is not the way you want to go. If you have polygons (plots) and year data (points), and you want the attributes from each year to become attributes of a single plot polygon, that method won't work (without renaming the attribute fields of the points to be unique). Other Overlay operations, such as Identity, Union, or Intersect, wouldn't work either. They would allow you to get the plot name as an attribute of the points, but not the other way around.
If you can better describe the current and desired format of your data (perhaps with screenshots of tables or just text examples), I may be able to edit a solution into this answer. Keep in mind that whatever analysis you want to do will play a role in the best way to organize the data. You might want to take a look at another question here: organizing attribute table: multiple sets of variables per point