No. If you lose the bag the contents are lost with it. Test it yourself, go into a cheat world spawn two canvas bags and you can put one cobblestone in one bag and then look in the second and it is not there. It is simply a means of increasing your inventory space.
If you're looking for something with the static storage/secure when you fall helplessly into a pit of lava, the Ender Pouch may work for you. It's not any where nar the size of an Alchemy Bag but it does have the ability to keep the static storage area.
Can I rename it? (Call me Nodai)
If you're willing to go into creative, use an enchanted book and the bag on an anvil. you can rename it and take it out. It won't actually have the enchantment but it will glow and have the name you chose.
Yeah, you don't need a book. That's stupid, why would you need a book just to rename an item?
What do you think?