Morton's Tender Quick is a fast cure salt meant to be mixed in with ground meats or used for curing thinner cuts (or fish). Hams must be brined for days in order for the salt solution to penetrate deeply through the flesh and prevent spoilage. It is important to follow good curing protocol and use the amount of salt as requested in the recipe.
Sodium nitrate, or pink salt #2, is not essential from a food safety perspective for whole muscle curing but is for dry-curing various salumi.
I suspect that this is a question that it's impossible to give a definitive answer for.
In reality it's probably a mixture of religion, culture and confusion.
I reckon that in most cases that it boils down to "Fish isn't a meat because when I was growing up I was told it's not a meat", or something like that.
In terms of etymology, "meat" originally just meant "food" and as such could be used for food of any kind. This carries over to some extent in modern usage - we sometimes talk about meat of a fruit/vegetable to describe the inside of it, coconut for example.
Personally I use meat to describe animal flesh regardless of the origin, I don't view the meat/fish deviation as being one that is either meaningful nor useful.
RE: fish and vegetarianism,
Fish isn't vegetarian.
Phew, glad we could clear that up.
There's an accepted meaning for the word and it excludes the eating of fish. There are a lot of people who eat fish and are otherwise vegetarian*, which is perfectly fine, but it's a misuse of the word. At the end of the day we all eat what we're comfortable eating, but it can be difficult grouping that many variations. This confuses others and in the end leads to inconvenience to vegetarians.
Oh language...
*Although this probably isn't true. If you're eating fish than it's obviously not on moral grounds so you're less likely to exclude animal products in some of the more obscure places, like may be in wines, cheeses, or indeed any processed goods.
Best Answer
I have had some success with this recipe: http://awesomepickle.com/pickled-herring-recipe-how-to-fillet-a-fish/
The fish should keep for a couple of weeks once pickled, but I always tend to eat mine in the first few days.