This is a common "gotcha" of the new syntax: case clauses are not expressions. That is, if you put a variable name in a case clause, the syntax assigns to that name rather than reading that name.

It's a common misconception to think of match as like switch in other languages: it is not, not even really close. switch cases are expressions which test for equality against the switch expression; conversely, match cases are structured patterns which unpack the match expression. It's really much more akin to generalized iterable unpacking. It asks the question: "does the structure of the match expression look like the structure of the case clause?", a very different question from what a switch statement asks.

For example:

t = 12.0
match t:
    case newvar: # This is equal to `newvar = t`
        print(f"bound a new variable called newvar: {newvar}")
        # prints "bound a new variable called newvar: 12.00000000"
        # this pattern matches anything at all, so all following cases never run

    case 13.0:
        print("found 13.0")

    case [a, b, c]: # matches an iterable with exactly 3 elements,
        # and *assigns* those elements to the variables `a`, `b` and `c`
        print(f"found an iterable of length exactly 3.")
        print(f"these are the values in the iterable: {a} {b} {c}")

    case [*_]:
        print("found some sort of iterable, but it's definitely")
        print("not of length 3, because that already matched earlier")

    case my_fancy_type(): # match statement magic: this is how to type check!
        print(f"variable t = {t} is of type {my_fancy_type}")

    case _:
        print("no match")

So what your OP actually does is kinda like this:

t = 12.0
tt = type(t) # float obviously
match tt:

    case int: # assigns to int! `int = tt`, overwriting the builtin
       print(f"the value of int: {int}")
       # output: "the value of int: <class 'float'>"
       print(int == float) # output: True (!!!!!!!!)
       # In order to get the original builtin type, you'd have to do
       # something like `from builtins import int as int2`

    case float: # assigns to float, in this case the no-op `float = float`
        # in fact this clause is identical to the previous clause:
        # match anything and bind the match to its new name
        print(f"match anything and bind it to name 'float': {float}")
        # never prints, because we already matched the first case

    case float(): # since this isn't a variable name, no assignment happens.
        # under the hood, this equates to an `isinstance` check. 
        # `float` is not an instance of itself, so this wouldn't match.
        print(f"tt: {tt} is an instance of float") # never prints
        # of course, this case never executes anyways because the
        # first case matches anything, skipping all following cases

Frankly, I'm not entirely sure how the under-the-hood instance check works, but it definitely works like the other answer says: by defintion of the match syntax, type checks are done like this:

match instance:
    case type():
        print(f"object {instance} is of type {type}!")

So we come back to where we started: case clauses are not expressions. As the PEP says, it's better to think of case clauses as kind of like function declarations, where we name the arguments to the function and possibly bind some default values to those newly-named arguments. But we never, ever read existing variables in case clauses, only make new variables. (There's some other subtleties involved as well, for instance a dotted access doesn't count as a "variable" for this purpose, but this is complicated already, best to end this answer here.)

If you just pass a type directly, it will consider it to be a "name capture" rather than a "value capture." You can coerce it to use a value capture by importing the builtins module, and using a dotted notation to check for the type.

import builtins
from typing import Type


def main(type_: Type):
    match (type_):
        case builtins.str:  # it works with the dotted notation
            print(f"{type_} is a String")
        case builtins.int:
            print(f"{type_} is an Int")
        case _:
            print("\nhttps://en.meming.world/images/en/0/03/I%27ve_Never_Met_This_Man_In_My_Life.jpg")

if __name__ == "__main__":
    main(type("hello"))  # <class 'str'> is a String
    main(str)  # <class 'str'> is a String
    main(type(42))  # <class 'int'> is an Int
    main(int)  # <class 'int'> is an Int

To expand on what I said in comments: match introduces a value, but case introduces a pattern to match against. It is not an expression that is evaluated. In case the pattern represents a class, the stuff in the parentheses is not passed to a constructor, but is matched against attributes of the match value. Here is an illustrative example:

class Tree:
    def __init__(self, name):
        self.kind = name
    
def what_is(t):
    match t:
        case Tree(kind="oak"):
            return "oak"
        case Tree():
            return "tree"
        case _:
            return "shrug"

print(what_is(Tree(name="oak")))    # oak
print(what_is(Tree(name="birch")))  # tree
print(what_is(17))                  # shrug

Note here that outside case, Tree(kind="oak") would be an error:

TypeError: Tree.__init__() got an unexpected keyword argument 'kind'

And, conversely, case Tree(name="oak") would never match, since Tree instances in my example would not normally have an attribute named name.

This proves that case does not invoke the constructor, even if it looks like an instantiation.

EDIT: About your second error: you wrote case type(Tree()):, and got TypeError: type() accepts 0 positional sub-patterns (1 given). What happened here is this: case type(...) is, again, specifying a pattern. It is not evaluated as an expression. It says the match value needs to be of type type (i.e. be a class), and it has to have attributes in the parentheses. For example,

match Tree:
    case type(__init__=initializer):
        print("Tree is a type, and this is its initializer:")
        print(initializer)

This would match, and print something like

# => Tree is a type, and this is its initializer:
#    <function Tree.__init__ at 0x1098593a0>

The same case would not match an object of type Tree, only the type itself!

However, you can only use keyword arguments in this example. Positional arguments are only available if you define __match_args__, like the documentation says. The type type does not define __match_args__, and since it is an immutable built-in type, case type(subpattern, ...) (subpattern! not subexpression!), unlike case type(attribute=subpattern, ...), will always produce an error.