I took a quick look at this and you can look up this experiment in SRA: https://www.ncbi.nlm.nih.gov/sra?linkname=bioproject_sra_all&from_uid=286762

For each file, it says the sample name under "Library". It also says the run number, which corresponds to the run accession number on the EBI ENA page for this study (https://www.ebi.ac.uk/ena/data/view/PRJNA286762).

Doing this manually for all 199 fastq files will probably not be fun. I hope someone else knows a better way :)

You can find the information at NCBI. For example, the BioSample links for all samples from this project. Here is the BioSample link for the particular sample from your post.

edit: you can use the sent to -> file -> format "Full text" at the page I and abbey linked ( https://www.ncbi.nlm.nih.gov/biosample?LinkName=bioproject_biosample_all&from_uid=286762 ) to save the metadata for all records. Here are the first and last records:

1: MIMS Environmental/Metagenome sample from gut metagenome
Identifiers: BioSample: SAMN04388293; Sample name: AM9; SRA: SRS1239704
Organism: gut metagenome
Attributes:
    /collection date="2014-10"
    /environment biome="intestine environment (ENVO:2100002)"
    /environment feature="mammalia-associated habitat (ENVO:00009002)"
    /environment material="feces"
    /geographic location="USA:Midwest:Columbia, MO"
    /host="Equus ferus caballus"
    /latitude and longitude="38.9514 N 92.3283 W"
    /animal_ID="1"
    /sample material processing="Power Fecal"
    /environmental package="host-associated"
Description:
Keywords: GSC:MIxS;MIMS:4.0
Accession: SAMN04388293 ID: 4388293

[...]

200: DNA isolated from fecal samples of multiple vertebrate species Identifiers: BioSample: SAMN03769901; Sample name: Gastrointestinal microbiota (fecal sample DNA from multiple animal species) Organism: gut metagenome Attributes:
    /breed="fish, mouse, cat, dog, horse"
    /isolate="Fecal samples"
    /age="unknown"
    /development stage="adult"
    /sex="pooled male and female"
    /tissue="fecal sample"
    /disease="Healthy"
    /geographic location="USA"
    /health state="healthy"
    /sample type="fecal samples"
    /storage conditions="-80" Description: DNA isolated from fecal samples of fish, mouse, cat, dog and horse. Accession: SAMN03769901  ID: 3769901

Use the NCBI RunSelector. To do so, go to NCBI front page and enter "PRJNA286762", selecting SRA in the dropdown menu left of the query field. This page will pop up. Click "Send results to Run Selector" to get the next page. Use the greenish "+" button to select all 199 files, then press "RunInfo Table". It will download a text file with the metadata you need. It includes the prep kit the sample was made from, the sequencer, total reads, SRR ID etc. From there on, you can simply filter out what you need.