Yes, Biosynthesis can incorporate multiple different types of dyes within an oligonucleotide, even for every 5 to 10 bases. We can place various dye labels at the 5'-end, the 3'-end, or internal positions. Depending on the desired modification, specific modifications can be incorporated using phosphoramidite reagents at the synthesis time.
However, some dyes can only be conjugated post-synthetically using various conjugation chemistries, such as maleimide-based conjugation or click chemistry.
It is challenging to synthesize and analyze oligonucleotides with multiple dye incorporation in some cases. Also, some products can be insoluble, and too many incorporated fluorescein groups may quench themself if the modified oligonucleotide is not properly designed.