(I have looked more carefully at this theory of "universal characters" mentioned by Stanley and am updating this answer according to what I learned. All of this was contained in the answer by Stanley, I am just unpacking it for the sake of those like me and the OP who may be confused.)

The following material is in the paper *Young-diagrammatic methods for the representation theory of the classical groups of type $B_n$, $C_n$, $D_n$* by Koike and Terada, also in works by King (such as *Modification Rules and Products of Irreducible Representations of the Unitary, Orthogonal, and Symplectic Groups*, Journal of Mathematical Physics 12, 1588, 1971) and a very readable 1938 book by Murnaghan (*The Theory of Group Representations*).

$\DeclareMathOperator\Or{O}\DeclareMathOperator\U{U}$So functions $o_\lambda$ correspond to irreducible characters of $\Or(2N)$ only when $\ell(\lambda)\le N$. The set of such actual characters forms a basis for the space of symmetric functions on variables $\{x_1,x_1^{-1},\dotsc,x_N,x_N^{-1}\}$. However, the coefficients in the expansion of power sums in general depend on $N$.

On the other hand, partitions with $\ell(\lambda)>N$ do not define irreducible representations, so $o_\lambda$ is not an actual character. In that case they are replaced by another symmetric function, $o_\lambda\to o_{\widetilde{\lambda}}$ with a modified partition $\widetilde{\lambda}$.

When these functions are used, the large-$N$ expansions continue to hold for small $N$. In that sense they are 'universal'.

When a Schur function $s_\mu$ is decomposed in terms of $o_\lambda$, which corresponds to the branching rule of $\U(2N)\supset \Or(2N)$, one has $\ell(\lambda)\le \ell(\mu)$, so if $\ell(\mu)\le N$ only actual characters appear. This is not true if $N<\ell(\mu)\le 2N$, and then universal characters must also be used. This is not discussed in the classic book "Representation theory" by Fulton and Harris, for example, where only the case $\ell(\mu)\le N$ appears.

The modified partition $\widetilde{\lambda}$ is defined as follows. For $O(2N)$, let $m=2\ell(\lambda)-2N$. Then remove from the Young diagram of $\lambda$ a total of $m$ adjacent boxes, starting from the bottom of the first column and keeping always at the boundary of the diagram. In this way the changes to be implemented are, in sequence, $\lambda'_1\to\lambda'_2-1$, then $\lambda'_2\to\lambda'_3-1$, etc. until the procedures stops at some column $c$. If $m$ is too large and there are not enough boxes to accommodate this procedure, or if the remaining diagram is not a partition, then $o_\lambda=0$; otherwise $o_\lambda=(-1)^{c-1}o_{\widetilde{\lambda}}$.

For example, the universal decomposition for $p_4$ is
$$p_4=o_4-o_{31}+o_{211}-o_{1111}+o_0.$$
If $\lambda=(1,1,1,1)$ and $N=6$, we must remove $8-6=2$ boxes, in which case we get $c=1$ and $\widetilde{\lambda}=(1,1)$. Hence, for $O(6)$ we have $o_{1,1,1,1}=o_{1,1}$ and the universal relation reduces to $p_4=o_4-o_{31}+o_{211}-o_{11}+o_0$. When $N=4$, we must remove $8-4=4$ boxes from $\lambda=(1,1,1,1)$. We end up with no boxes at all, so $o_{1,1,1,1}=o_\emptyset$ for $O(4)$. We must remove $6-4=2$ boxes from $\lambda=(2,1,1)$, arriving at $o_{2,1,1}=o_2$. The relation reduces to $p_4=o_4-o_{31}+o_{2}$. Finally, take $N=2$. We cannot remove $8-2=6$ boxes from $(1,1,1,1)$, so $o_{1,1,1,1}=0$ for this group; when we remove $4-2=2$ boxes from $(3,1)$, the result is not the diagram of a partition, so $o_{3,1}=0$ for this group; removing $6-2=4$ boxes from $(2,1,1)$ leads to the empty partition, and the removing procedure ends in the second column so $c=2$, hence $o_{2,1,1}=-o_\emptyset$ for this group. Thereby the relation reduces to $p_4=o_4$.

Unfortunately, the modification rule is incorrectly stated in the recent "The Random Matrix Theory of the Classical Compact Groups" (Cambridge University Press, 2019), by Elizabeth Meckes.