Usually you will want determine if the observations are paired or not, based on the theoretical relationship of how the measurements were taken. That is, if two measurements are from the same respondent or experimental unit, you will want to consider them paired.

Paired data may not be correlated in the way you suggest. Consider the follow two sets of observation. These (made up !) data come from paired observations.

Time1 = c(1, 4, 3, 3, 3, 3, 4, 3, 3, 3)

Time2 = c(4, 5, 4, 5, 4, 5, 3, 4, 3, 4)

You notice that 8 out of 10 observations increase from Time1 to Time2. But they are not correlated.

For theoretical reasons, you would want to consider these values paired. (Even though, in this case, you would get a lower p-value treating them as unpaired observations.)

Being described as paired means the two measurements were done on a single individual. Where that is not true of a set of measurements they are not considered as paired. It is not usually based on the correlation of the pair.

Analyzing the data as paired or unpaired is an issue that depends on the design of how you intend to collect them.

If, before collecting the data, you have clear criteria for defining pairs, then you have paired data and you can analyze them as such, including calculating correlations or paired t-tests. If not, you would have to match them in order to apply these analyses, but it is almost certain that by doing so, you would introduce scientific and even ethical bias into your research. My recommendation is that you go to an advisor who knows the subject.