equals method for date comparison in the library java.util.Date

Is the method java.util.Date method .equals indifferent in what concerns the date format or not?

A Date object doesn't have a format - it has no notion of where the information came from. It is just a number of milliseconds since the Unix epoch (January 1st 1970, midnight UTC). It also doesn't know about time zones. So you could have two Date objects created in very different ways which end up representing the same point in time - and equals would return true there.

It's really important to understand what information is and isn't part of the state of an object - particularly for date and time APIs, where it can be counterintuitive in some places. When in any doubt, read the documentation carefully. For example, from the documentation for Date:

The class Date represents a specific instant in time, with millisecond precision.

and in equals:

Compares two dates for equality. The result is true if and only if the argument is not null and is a Date object that represents the same point in time, to the millisecond, as this object.

Thus, two Date objects are equal if and only if the getTime method returns the same long value for both.