This is the Adafruit BMP085/BMP180 library : GitHub - adafruit/Adafruit_BMP085_Unified: Unified sensor driver for Adafruit's BMP085 & BMP180 breakouts
I assume that you tried the example from that library and not your own sketch. Is there any error output from the example sketch ? The output that you show is not the output from that example sketch.
However, first the Pro Mini should be running normal. Perhaps you have a 3.3V 8MHz version with a 16MHz crystal. Can you try to read the crystal or resonator with a magnifier ? Sometimes there is vaguely some text that starts with 8... or 16...
Can you be more precise ?
Did you select "Arduino Pro or Pro Mini" and "ATmega328 (3.3V, 8MHz)" ?
And after uploading, the sketch runs twice as fast and the baudrate is twice as high ? Then it must be a 16MHz crystal.
Are you sure the voltage regulator creates 3.3V and not 5V ? Perhaps it is a 5V 16MHz Pro Mini after all.
You could use it as a 5V 16MHz Pro Mini and ignore the fact that it runs at 3.3V. However running a ATmega328P at 3.3V with 16MHz is outside the specifications. You better buy a new Pro Mini from another supplier.