There's various ways to do this in Modelbuilder and scripting, but I'll outline what I think is probably the easiest conceptually and practically for someone new to modebuilder or python scripting. i.e. this is not how I would do it but it will make the most sense to you.
In Model Builder you would first iterate of selections of the same species within the feature class using https://pro.arcgis.com/en/pro-app/latest/tool-reference/modelbuilder-toolbox/iterate-feature-selecti...
then for each selection, create a buffer https://pro.arcgis.com/en/pro-app/latest/tool-reference/analysis/buffer.htm of 50 metres.
you would then have 2 tools connected to the new buffer output FC - dissolve https://pro.arcgis.com/en/pro-app/latest/tool-reference/data-management/dissolve.htm and intersect https://pro.arcgis.com/en/pro-app/latest/tool-reference/analysis/intersect.htm
the intersected output would be an area where at least 2 of the same species are within 100m of each-other. NB I'd need to better understand how you want the territory defined, as this may not be what you want.
Time - I would first do a spatial join https://pro.arcgis.com/en/pro-app/latest/tool-reference/analysis/spatial-join.htm of the points against the dissolved output, using a Merge rule 'Join' of the date field with a delimiter such as a comma.
Then I would do the same spatial join with the intersected output previously. This resulting feature will then have the intersection of sightings for a single bird species (remember this is in an iteration so when this is done, the process is automatically repeated for the next species) and a field of comma-separated dates. I would then use calculate field and a python calculation to create a set() from the dates, and if the set length (len(set) > 1) return something like 'true'.
Thus any features where this field is 'true' would indicate a territory. If you want a point, just use feature centroid to point.
I appreciate this is not simple for someone new to this, so I'd recommend breaking this down into bitesize chunks, and of-course there may be something simple I'm missing which someone else can point you to.
Also recommend figuring out what you envisage the territory to look like, as this will affect the logical steps in the model.