QGIS same print layout for multiple features of a layer?

The QGIS Atlas does not require that each page in the series be spatially distinct. In the map frame's Item Properties, ensure that the option Controlled by Atlas is not checked.

It does seem, however, that you have a series of attribute values, as opposed to individual features, that you want an atlas for. This can still be accomplished using an atlas, but will require a separate layer. You can easily use a standalone table for this.

Here is an example with a lines layer Lines, which has a type attribute. By using the built-in tool List Unique Values, you can generate a standalone table on your type field

In your Atlas Settings, choose the standalone table as your coverage layer, with the type attribute as the page name.

On the Lines layer, change the symbology to Rule-based. Define a single rule, using the expression "type" = attribute(@atlas_feature, 'type')

Now you can generate your atlas!

I would create an atlas with a geom for each species (all geoms should be the same, a just a point in the same place or a boundary that defines the extent) and each geom should have an attribute with the species name. Set the atlas page name to be the atlas layer attribute. You can then use @atlas_pagename as the title and filename of the export of each page

Then set the symbology of the actual species layer to only show the data where the species name attribute matches the atlas pagename.

You can then export as a single atlas.

what might be easier if you don't have too many species is to manually create different layouts for each species and export all of them at once with the plugin Maps Printer.