I found the following passages in “A Course in Universal Algebra” by Burris and Sankappanavar.
One cannot hope to find any further essentially new lattice properties which hold for the class of lattices of subuniverses since every algebraic lattice is isomorphic to the lattice of subuniverses of some algebra.
from “II§3 Algebraic Lattices and Subuniverses” and
One cannot hope for a further sharpening of the abstract characterization of congruence lattices of algebras in 5.5 because in 1963 Grätzer and Schmidt proved that for every algebraic lattice $\mathbf{L}$ there is an algebra $\mathbf{A}$ such that $\mathbf{L}\cong\operatorname{\mathbf{Con}}\mathbf{A}$.
from “II§5 Congruences and Quotient Algebras.”
I understand that they belong to a meta-level of mathematical reasoning. I'm struggling to understand their meaning and why they are true. Of course, I'm referring to the first parts of every passage, not the formal facts about an isomorphism in the second parts. I think that a lattice property is a (logical) formula involving lattice operations.
The authors simply mean that you will never find a lattice theoretic property that is true of all lattices of subuniverses (or all lattices of congruence relations) but not true of all algebraic lattices.
The next two paragraphs simply reiterate this in a less concise way, but might help to clarify. After that I'll point out a class of lattices for which the situation is (or might be) different.
Suppose we find some property P that is satisfied by every congruence lattice and suppose we believe P might help us characterize such lattices, i.e., distinguish them from other (non-congruence) algebraic lattices. Then we should be able to find at least one algebraic lattice that does not satisfy P (otherwise the property is true of all algebraic lattices and therefore doesn't help characterize congruence lattices).
The Gratzer-Schmidt Theorem assures us that we'll never find such a P. The theorem says that every algebraic lattice is (isomorphic to) the congruence lattice of some algebra. So every lattice theoretic property that is true of all congruence lattices is also true of all algebraic lattices.
Contrast this with the (current) situation for congruence lattices of finite algebras. Since it is not yet known whether every finite lattice is the congruence lattice of a finite algebra, there is still hope that someone will someday find a property that is true of all congruence lattices of finite algebras but not true of all finite lattices.
Here's one possibility
(P) the property of being not isomorphic to the following lattice:
It's possible that all congruence lattices of finite algebras have property P. (see also this MO question)