I would say “Aujourd’hui c’est lundi. / C’est lundi.” and “On est lundi.” are the two best way of saying it.

“Aujourd’hui c’est lundi. / C’est lundi.” is the most formal way and “On est lundi.” is a bit less formal, but still very appropriate.

I personally never used “Nous sommes lundi.” and don’t recall ever hearing or even reading it.

Nous sommes lundi is a slightly formal/written way to tell which day of the week it is. One can also say Nous sommes [le] lundi premier septembre.

This French idiom can only be used in the first person plural.
You can’t say Je suis lundi or ils sont lundi because the current weekday is expected to be the same for everyone.

The same idiom exists in Spanish: Estamos a lunes and Italian: Siamo [a] lunedì.

In English, while “We are Monday” will sound very foreign, one can say “We are late“, “We are on time” or “We are in September” which are close.

Aujourd’hui, c’est lundi or simply c’est lundi are fully acceptable equivalents.

On est lundi is a less formal and more common way to say nous sommes lundi.

Think of

Nous sommes lundi.

as

We are monday.

(which in fact is the word for word translation as you already know)
as in

We are on a monday day and on this day we do laundry.

Both

Aujourd’hui c’est lundi. / C’est lundi.

and

On est lundi.

are correct as well. And the last one is indeed less formal as it uses on instead of nous

A possible explanation might be the underlying idea “Nous sommes arrivés à lundi (dans la série des jours de la semaine).