The dot product of to vectors $v$ and $w$ is often defined as $dot(v, w) = v^T w$. This is usually done in the context of matrix representations and assuming that a vector is a column matrix. However, if we think of vectors in more abstract terms, we don't necessarily have to define a basis, let alone think of them as lists of numbers or column matrices. Does it even make sense to talk about the "transpose of a vector" in this abstract way of thinking?
At most, I can think of the transpose of a "linear transformation" (that could, but doesn't have to, be represented as a matrix) $M$, which we could define as an operation that satisfies the following rule: $dot(Mv, w) = dot(v, M^T w)$, which I think is a definition that has some high-level meaning if we think of $M v$ as the result of applying the linear transformation $M$ on $v$, and the dot product as a function defined on two vectors; we never had to refer to matrices or lists of numbers. However, does the "transpose of a vector" have any useful high-level interpretation (does it yield a new vector? a new linear transformation?) or is it just a convenient notation that only makes sense if we assume that we are working with matrix representations?
Given a vector space $V$ over the field $K$, the dual space $V^*$ consists of linear functionals $\alpha:V\to K$. Given a basis $\{e^1,\dots,e^n\}$ of $V$, its dual basis is a basis $\{\varepsilon_1,\dots,\varepsilon_n\}$ of $V^*$ such that $\varepsilon_i[e^j]=\delta_{ij}$. If you represent elements of $V^*$ as row vectors, then, relative to a pair of dual bases, applying an element of $V^*$ to a vector in $V$ becomes matrix multiplication. You can then interpret the transpose operation in $v^Tw$ as a linear map that sends basis vectors of $V$ to their duals.
The transpose of a matrix also has a connection to dual spaces. The adjoint $T^*$ of a linear operator $T:V\to W$ is the linear map $T^*:W^*\to V^*$ (note the reversal of direction) such that for $\beta\in W^*$ and $v\in V$, $(T^*\beta)[v]=\beta[Tv]$. It turns out that if $M$ is the matrix of $T$ relative to a pair of bases for $V$ and $W$, then $M^T$ is the matrix of $T^*$ relative to their duals.
(If the underlying field is $\mathbb C$, then you’ll need to use conjugate transposes instead.)