We usually use "eis" in Portuguese while telling jokes or imitating someone! Like, "eis a questão". I've particularly never seen this word used in another context in conversations!

They are almost the same, but in informal conversations it would be better use "aqui está". Hardly ever you will hear someone saying "eis". Hope this helps!

"Eis" could be related to realise something, per example, in Hamlet by Shakespeare you will find the phrase "...Be or not to be, thats the question..." that in Portuguese you will find "Ser ou nao ser, eis a questao" which means he realise him self in a situation that a decision should be taken; "Aqui esta" is related to show something near or to catch directly attention to something; Hope it could help giving some idea.

"Eis" é um advérbio com o verbo assimilado. O seu significado é mais sublime do que "aqui está". "Aqui está" é neutro, "eis" definitivamente quer chamar a atenção e importância para o fato. A prova disso é que podemos dizer: "Eis" is an adverb with the verb assimilated. Its meaning is more sublime than "aqui está". "Aqui está" is neutral, "eis" definitely wants to draw attention and importance to the fact. The proof of this is that we can say: "Eis que aqui está..." Então "eis é uma mistura entre" "aqui está´", "reparem bem ", "contemplem". En inglês é traduzido muitas vezes por "behold" e em francês por "voilá". So "eis" is a mix among "here it is", "pay attention", "contemplate". In English it is often translated by "lo, behold" and in French by "voilá". E eis que os soluços desapareceram. => And lo and behold, the hiccups went away. E eis que Davi não pecou mais. => And behold, it came to pass that David sinned no more. Então, eis o aspecto do nosso planeta daqui a vinte anos. => So, this is what the planet's going to look like in 20 years. Eis senão quando (idiomatic expression)=suddenly

"Eis" is a funny way to say "aqui está". You can use any of them, but it is most common to say "aqui está".