Proof for Turing Machines being able to simulate any algorithm in the same time complexity
This is not a formal statement, but either a definition or an empirical observation. The full claim is something like the following:

"A reasonable model of computation (for discrete data) together with a reasonable notion of time complexity will have the property that it can be simulated by a Turing machine with at most polynomial increase in time complexity."

The two occurances of "reasonable" here are generally not seen as formal notions, but rather as informal notions. We can trivially define models of computation with notions of time complexity not having the property above, but these models do appear artificial. On the other hand, models people actually want to use have the property.

Any attempt to formally define what a "reasonable model of computation with time complexity" is would probably include the possibility of polytime-simulation by a TM as an axiom.