yes, this is possible. in the Layer Options, there’s a Size slider that will allow you to adjust the size of the markers. you can also change other marker properties, like color, opacity, etc.
i didn’t add a genus column to the data, but below is an example Vienna Street Trees map in ArcGIS Online (https://www.arcgis.com/apps/mapviewer/index.html?webmap=b1683a8638dc42499645a093dff708aa). to filter by a particular species, you could select Layers > Vienna Street Trees > Options > Show Properties… and then on the properties bar on the right, use the Filter tool to filter by species, as shown in the screenshot below:
the source data is an abridged version of the government data, stored here: https://www.arcgis.com/sharing/rest/content/items/afd55b6a4c7745419e8b753914e61dc2/data.
note that i’m just providing the map and data here as an example for now, and i may delete them at a future point in time. if you want to to reference this in the future, it’s probably best for you to save the underlying data to your own shared location and create your own map. (it’s only a 2-layer map. so it shouldn’t be much work to replicate.) it’s worth noting that i tried pointing the map directly to the Vienna government link you provided, but for me, it responds too slowly or else the file is too large, and ArcGIS Online times out when trying to access that data. (that’s why i created my own abridged set of the data. it retains all the rows but eliminates a lot of columns just for space savings. on the map, you can click the points, and it will show only species and lat/long because those are the only fields that i’ve retained in my data set.)