I think one of the reasons you struggle is that clustering DNA sequences is not a clearly defined task. In general intention of clustering is to reconstruct, or approximate the relatidness of the DNA seqeunces.
If all these sequences are homologous you can do a multiple sequence alignment (using MAFFT or Clustal) and build a phylogenetic tree (using something like IQ tree). If your prof suggested ClustalW, this is probably what they meant. Evolutionary speaking, that's the optimal method for getting relations of sequences, but they must be "alignable". Also with this many sequences, it will be computationally quite heavy.
For within species relatedness, phylogenetic reconstruction is sometimes impossible, because recombination breaks down the linkage of individual alleles. However, if all of the sequences are long enough, you can do a principal component analysis on SNPs, for example using something like plink. The plot you showed actually looks a lot like PCA (it even has the explained variability on axes).
Finally, regardless of their relationship, you can always do a dendrogram based on a similarity matrix, but for that, you need to define similarity. That will be usually something kmer-based. I am not following this field very closely lately, but there are some recent papers about kmer-based clusterings, like this one.
When looking for the paper I also run into this one, which proposes genome clustering using "Inter-nucleotide Covariance". No idea how useful/sound it is. Just found the abstract quite interesting.
Long story short, it really depends on what you are after and what are your sequences! If you can build a reliable tree, it is the most powerful way to determine the relationships among sequences.