[inglor_cz]

Oaths tend to be sworn to people with a deity as a guarantor and with the understanding that the invoked deity will punish you accordingly if you break your oath.

Without the "deity as a guarantor" element, the oath becomes a bit weird. The substitution of, say, the US government for the deity would probably work in the narrow sense (the government will, after all, probably retribute painfully if you break your oath - at least for oaths that matter to them), but then again the religious dimension becomes awkward, because public servants are mere mortals like you.

Bret Devereaux has a useful blog article on this topic:

https://acoup.blog/2019/06/28/collections-oaths-how-do-they-...

As with many other rituals inherited from a distant past, there is a discrepancy between what we do and what we believe. In the times when oaths were first introduced, open atheism would be extremely rare. Nowadays, it isn't, so the original construction starts to come apart at the seams.

[mindslight]

It only "comes apart at the seams" for people that believe there is some ambient authority, which then becomes less powerful or disinterested due to not being involved with the oath.

The secular view is that the person swearing is pointing to a symbol and promising to uphold the values it represents, regardless of whether that symbol represents a traditional organized religion or not.

[azernik]

Though there is, as I commented there, in which the "on penalty of perjury" construction very closely mirrors the original religious structure, with punishment by the state taking the place of punishment by God or the gods. Interestingly, lots of states require the "on penalty of perjury" only in cases where a religious oath is not being made.