Yes you can integrate your chatbot built on FB Messenger Platform on your own website :)
Integrating your FB Messenger chatbot onto your website
With FB's latest Messenger update, it is now possible to embed your chatbot onto your website so that users can interact with your chatbot while browsing your website. The full feature & implementation details can be found in FB's documentation here. Here's a nice tutorial on how to do it here too. However, you still need to have a FB page linked to your FB Messenger bot to do so because bots are mainly used for talking to users via your FB page.
Can FB Messenger Platform be integrated to a different platform?
Well, no, but various other platforms such as API.ai (DialogFlow) or WIT.ai have integrations with other messenger platforms such as Telegram, Slack, Kik, etc.
I hope this helps! I recommend API.ai (DialogFlow) as a platform for making chatbots since it's relatively easy to learn and has good documentation. If you want a quick way to prototype a chatbot, then I recommend using ChatFuel, before making one from scratch :)