The relation $\leq_{\cal U}$ is very well-studied: it is the order relation on the model of the hyperreals obtained by taking an ultrapower of $\mathbb{R}$ with respect to $\cal U$.
Letting $\mathbb{R}^*_{\cal U}$ denote this ultrapower, you are asking for the smallest size of an unbounded set in $\mathbb{R}^*_{\cal U}$ (which you call $\frak b_{\cal U}$) and the smallest size of a cofinal set in $\mathbb{R}^*_{\cal U}$ (which you call $\frak d_{\cal U}$).
As Wojowu points out, these cardinals are the same because $\leq_{\cal U}$ is a linear order on $\mathbb{R}^*_{\cal U}$. Also, this cardinal is bounded below by $\frak b$ and above by $\frak d$.
But one can say a bit more. Andreas Blass has proved (in this paper) that $\frak g \leq \frak b_{\cal U}$. Mike Canjar proved that, if you add lots of Cohen reals to a model of CH, then any uncountable regular cardinal $\leq \frak c$ is equal to $\frak b_{\cal U}$ for some free ultrafilter $\cal U$. In other words, he showed that the value of $\frak b_{\cal U}$ can depend not only on your model of set theory, but within a single model it can also depend on $\cal U$.
If you want to learn more, I suggest you look at Andreas's paper, and also his answer to this related questionthis related question of Joel David Hamkins.
