It's hard to translate this phrase into English, because English has only one form for 'you', regardless of whether one person or number of people are being addressed. In Russian, 'вы' (plural) is used to address a group of people (something like 'you guys'), and 'ты' (singular) to address a single person. Also, Вы is used to address a person in a polite or/and formal way (a stranger, person older than you, business partner etc.). So, if somebody says to you 'давай перейдем на ты' (let's switch to ты) or, as in your example, 'можно и на ты', it means that this person doen't want to be too formal.

I would translate it as "If you want you can feel more informal with me" Literally "со мной можно и на ты" = Feel free to call me "Ты". The matter is that in Russian language there are 2 translations of English "you". Here they are: 1."Ты" = informal "you" 2."Вы" is formal "you" Sometimes your new acquaintance may suggest you to use "Ты" while talking to him or her. It's a friendly gesture. :)

In russian language we use do not make any difference between for pronoun "you" in single or plural from. It means - "you" or "you". When we traslate from english into russian to show politeness we translate as "you". I consider you chatted to someone in russian and said "You." Your friend proposed you to say - And you can say me "you" if you want.

By the way, you can address me by "ты" (opposing to "вы")

He or she suggests you to use "ты" rather than "Вы" when chatting. In English you say "let's get on a first-name basis," right?