Calorie goal isn't correct

It may be a problem with my reading comprehension, not what you wrote, but I admit I'm not completely clear about all aspects of what you're saying.

When you say you told MFP you want to lose 5 pounds, are you talking about the goal weight setting? That has no effect on calorie goal.

What affects calorie goal - besides your demographic data and activity level - is the weight loss/gain rate you told MFP you wanted, or the maintenance setting that's an option in that same area. If you told MFP you want to lose half a pound a week, it subtracts 250 calories from what it estimates you'd burn in a day, and makes the result your goal. Obviously, it subtracts more if you said you want to lose faster, adds calories if you said you want to gain, and makes its estimate of your calorie burn your goal. However, MFP will never give a man a goal lower than 1500, or a woman a goal lower than 1200, so people asking for unsafely fast weight loss may not get the expected deficit.

Main point: What affects your calorie goal is whether you said you wanted to gain, maintain or lose weight, and in the case of gain/lose, how fast you indicated you'd like to do it. Goal weight is not relevant in that calculation.

Conceptually, by the end of the day, the tracker sync should reconcile what MFP expected a person to burn with what the tracker actually saw them as burning, and MFP will keep any requested deficit/surplus in the picture arithmetically. However, for that to be fully operative, negative adjustments must be turned on in MFP. Without negative adjustments turned on, MFP will add calories when the tracker says the person burned more calories than MFP expected, but won't subtract calories when the tracker says the person burned fewer calories than MFP expected.

Do you have negative adjustments turned on?

The goal may or may not be accurate, even with a tracker sync: It's just a starting estimate, an average for demographically similar people. A 4-6 week trial (or one whole menstrual cycle for those who have them) should provide feedback about whether an individual is close to the averages on which MFP and trackers base their estimates. Most people are close, but a few aren't. Even trackers just estimate based on people of similar demographics, they just do it in a more detailed, nuanced way. Garmin is off by several hundred calories daily for me, compared with nearly 10 years of successfully - and quite predictably - losing/maintaining weight by logging in MFP. That's rare, but possible.

In general, IME the 5-week prediction is just silly-wrong, so I wouldn't put much emphasis on that.