Why does the Duplo bathtub piece have an odd height?
Although we probably won't get a definite answer to why this decision was made, unless a LEGO part designer steps in to answer the question, here are some things to consider.

The LEGO DUPLO bathtub, toilet, and sink all have two studs at the end. Looking through all the sets these pieces appear in, the studs are used for attaching accessories, like a shower head (for the bathtub), mirror (for the sink), or printed bricks with various bathroom accessories. They are not made to be integrated into a wall (at least not as a primary feature).

Even though the toilet and sink are 1.5 LEGO DUPLO bricks tall with two studs, I can find very few examples when this was taken advantage of by incorporating the pieces into a wall. In the vast majority of LEGO DUPLO sets, the bathtub, toilet, and sink are freestanding next to a wall, and often next to each other. To mimic realism, the bathtub needed to be shorter than the sink, but tall enough to fit a figure - thus the odd height. 

In the case of LEGO DUPLO accessories, furniture pieces, and fixtures, this is often the case. LEGO DUPLO element designers will make sure a piece can fit with other pieces in at least one primary way, but there are other considerations as well, like scale in relation to other pieces and fitting the figures properly. 

We should remember that while LEGO DUPLO is a construction toy, it is meant to be for the youngest of children. So, while building and rebuilding is possible, the parts are not designed to be attached using complex building techniques or repurposed in many different ways, as we see in LEGO System sets. 

As far as the idea of integrating the bathtub with the slide, how about putting a couple of LEGO System plates under the pieces to even out the height? LEGO System plates are compatible with LEGO DUPLO if placed on the underside. I do not have the pieces to test this out myself, but OP's question made me curious.