It’s more complicated than that even, as some states amend the NEC to change the rules, for example here in MA.
The 2020 NEC (NFPA 70) 334.80 paragraph 1 reads:
334.80 Ampacity. The ampacity of Types NM and NMC cable shall be determined in accordance with 310.14. The ampacity shall not exceed that of a 60°C (140°F) rated conductor. The 90°C (194°F) rating shall be permitted to be used for ampacity adjustment and correction calculations, provided the final calculated ampacity does not exceed that of a 60°C (140°F) rated conductor. The ampacity of Types NM and NMC cable installed in cable trays shall be determined in accordance with 392.80 (A).
Which is very clear: you use the 60C column, and so copper #6 is good for 55*.8=44A continuous.
But here in MA (
https://www.mass.gov/doc/state-electrical-code-massachusetts-amendments-2020/download ) they modify this section:
334.80. Delete the second paragraph and revise the first paragraph to read as follows:
334.80 Ampacity. Type NM, and NMC cables shall have conductors rated at 90°C (194°F). Where installed in thermal insulation, the ampacity of conductors shall be that of 60°C (140°F) conductors. The ampacity of Types NM, and NMC cables installed in cable tray shall be determined in accordance with 392.11.
With this updated wording, NM (romex) is only limited to the 60C column if it is installed in insulation. Otherwise, you can treat it the same as THHN/90C rated wire (which makes sense, because… it is the same!). That means for most cases, using the 75C column, giving you 60A*.8=48A continuous.
Also, in either case, a 60A breaker is the correct size breaker for #6 NM/romex
TLDR: If your #6 romex is not installed in insulation, it is perfectly safe to run at 48A continuous. Whether or not that meets your local code depends on the specific local language that was adopted.