Is adaptive security the same as full security? This question contains a great answer about adaptive security, but I haven't yet found a proper definition what full security is. Does a stronger security than adaptive security exist or are the two synonymous.

If they are different, what is the difference? If they are not different, why are they referred to differently?
I don't know what you mean by "full security", but "adaptive" just means that the oracle queries from the attacker depend on the results of previous queries. In contrast, "non adaptive" means that the attacker simply makes all the queries in batch, that is, without adapting his queries from previous results.

Update: based on the paper from your comments, it seems that "full security" is opposed to "selective security". Both "full" and "selective" security are defined in Chapter 2.3.3 of that paper. According to that definition, both models are adaptive, since queries from Phases 1 and 2 are adaptive.