In a sense, yes. But I think we is even less informal than the standard contractions in English. Otherwise the restrictions in usage seem to be very similar. E.g. is not and isn't aren't fully interchangeable, either. You can't use isn't when you want to stress not (as there is no vowel to stress in n't) any more than you can use we when you want to stress wij. But in most situations you have a free choice, and picking one or the other doesn't really mean anything.
Eten is the plural conjugated form of the verb (wij/jullie/zij eten = we/you(pl)/they eat), the infinitive (eten of niet eten = to eat or not to eat) and the present participle (ik hou van eten = I like eating). These forms are the same for all Dutch verbs.
Eet is the singular conjugated form of the verb (ik/jij/hij/zij eet = I/you(sg)/he/she eat(s)). Normally the second and third person would be indicated by adding -t at the end just like in English the third person is indicated by adding -s, but since eet already ends with a t this is not done.
Eet vs eten is sort of like the difference between eat and eats in English. Eet is for ik, jij and hij (i.e. singular), and eten is for wij, jullie and zij (i.e. plural).
In a chart that's:
- ik eet
- jij/u eet
- hij/zij/het eet
- wij eten
- jullie eten
- zij eten
"We eet" would be just as wrong as saying "We eats" in English.
Use eet if one person eats (singular), eten if more than one person (plural):
- Ik eet, jij eet, hij/zij/het eet.
- Wij eten, jullie eten, zij eten.
Eten (this is also the infinitive) is even simpler than most other Dutch verbs because it already ends with a t, so third person singular is just like first person singular.