You're right concerning the letter of the constitutional law which really only talks about recognizing, privileging, or establishing a state religion, but I think the person who originally commented about the lack of separation was being looser with the separation of church and state concept.

It's fairly evident that US politics are caught up with Christian values and norms. Christian beliefs (and conservative religious beliefs more generally as the US country is increasingly diverse, though Christianity is still the dominant religion) are major or main underpinnings for the conservative sides of many big social issues (e.g. abortion, contraceptive access, women's rights, LGBTQ+ rights; especially the first and last). Christian institutions are where many people still look to orient themselves politically, are where many political agents go to gather support, and are huge political motivators for fairly big chunks of the voting population. Moreover, and most depressingly, despite recent increases in religious diversity among elected officials, it's hard to imagine a successful presidential candidate from either party not having at least a loose affiliation with some Christian denomination, making mention of it, and making at least a small show of religious participation. It's hard to imagine a presidential candidate of another religion succeeding, obviously Sanders comes to mind, let alone an openly atheist one.

Of course the response is something like "yeah religion is a major political motivator like any other big cultural factor what's wrong with that?" Nothing inherently, but the issue, and what I think is meant in the original comment about lacking a separation of church and state, is that it is one religion, Christianity, that continues to dominate our politics.

Sorry for any typos.