It's not about X, it's about Y.

Joe Biden recently said:

'This is not about freedom or personal choice, it's about protecting yourself and those around you.'

Another example would be, in the case of increased surveillance:

'It's not about privacy or the freedom to go about without being watched by the government, it's about stopping the terrorists.'

The statement does not establish any reason why freedom, or another value, should now be sacrified to achive some other goal, it simply assumes it the new goal overides any other consideration. In this way I feel like it's sort of begging the question, but maybe there is another name for it.

This is more of talking point than an argument. As such, it is not meant to be rational or logical rather it is meant to affect people emotionally. In that sense, we might call it an appeal to emotion — especially if the statement is made in isolation and not explained (the logic part). Keep in mind that effective persuasion involves both reason and emotion. So if the reason part is there, there is nothing wrong with using emotion too (just not in place of reason).

 An unsupported claim. The implicit argument is likely:

P1: In a national emergency such as war and pandemic, the right to privacy has often had to accommodate national security and the mutual welfare." (Insert historical examples)

P2: Covid 19 is causing nationwide sickness and deaths

P3:  Nationwide sickness and death is a national emergency (examples: stress on the healthcare system, etc.)

C:   Therefore, the right of privacy and individual freedom should accommodate our national security and mutual welfare.

'This is not about freedom or personal choice, it's about protecting yourself and those around you.'

Could e a opinion. The words freedom and person choice are really subjective terms.

Could also be an appeal to emotions. if you do not do this than you are hurting the people around you.

The phrase can be applied to any number of things. This is what I call bumper sticker logic.

Off the top, the statements seem like, well ... statements, not arguments.  As such, they may reflect personal opinions and not logical arguments.

Viewed from the perspective of an argument, and taken purely as the words read, each statement seems false.  Whatever the "this" is, it seems to be about ALL of: freedom, personal choice, protecting one's self, and protecting others – not one or the other alone.  However, it's important to accept that not all 4 of those elements weigh equally for a decision and it's important to accept that there may well be other elements to consider before deciding which particular course of action is the best.

If the goal is to decide on a course of action, it will come down to whether  personal choice is more important than protecting others.  For those who believe protecting others is of supreme importance, there's one "best" option; for those who believe personal choice is of supreme important, there's a different "best" option.  A decision about which course of action to take will come down to the influence the one considers most significant.  That's a subjective, value judgement ... not one that will get resolved through the use of logic.  Which dress is prettier ... the red one or the green one?  Logic won't help us answer that question, either.