How can I verify that all possible FLOAT(53) values convert to unique BINARY(8) values?
The size, in bytes, of both BIGINT and FLOAT(53) are documented.  It would be a breaking change to make either one not convertible back and forth to BINARY(8).

The documentation you quote is about the details of the conversion, which are undocumented and subject to change.  EG the byte layout of a float(53) in this conversion could change.  So if you convert a float to binary(8), store the binary(8) in a table, upgrade SQL Server, and convert the binary(8) back to float, you might not get the same value.

These particular types have well-known byte layouts.  BIGINT is an 8-byte little endian integer, and float(53) is a standard double-precision floating point number.  And it would be pretty surprising if one of those changed.  But other types, especially CLR-based types, really might change their layout between versions.