I recently bought this solder paste, thinking solder paste should do the trick when not using tin wire. Did I buy the wrong solder paste or does solder paste have another function since looking at it it doesn't seem to produce a connection with tin?
There is flux paste and there is solder paste. Flux paste is just flux, no metal solder. Solder paste is metal solder beads suspended in flux for use on SMD components in reflow.
It could be that your solder paste just has really fine solder beads. I've never handled solder paste so I don't know how fine it can get.
Are your components are firmly stuck to the board and conductive?? If you can get an eye loupe or microscope you might actually be able to see it. Stick multimeter probes into your solder paste and measure the resistance.
EDIT: Some amazon reviews say it is not labelled correctly and is actually flux paste.
This "solder paste" is not what is typically called solder paste. Solder paste, which is used for reflow soldering, is a mixture of tiny alloy particles (Sn-Pb, Sn-Sb or Sn-Ag) and flux. Your paste is bare flux and contains no solder. A better translation for the german word "Lötfett" would be "flux paste".
For reflow soldering, this kind of flux is usually not suitable because it typically contains acidic incredients which are corrosive and lead to leak currents on the pcb surface. It should only be used for soldering wires and non-electronic soldering.
Look for a solder paste that has the alloy specified on the package.