Which plastic type is used inside dishwashers?
One possible reason for the different behaviour of the drinking bottle, compared to the lining of the dishwasher may be indeed the different composition of the polypropylene deployed.  The part of poly indicates that many hundreds and thousands smaller propylene molecules are brought together to constitute the polypropylene.  Yet there are different ways this may happen, as this picture (source) depicts:



Along the polymer chain, there a methyl groups (marked as $\ce{CH3}$) occupying more volume than hydrogen atoms (marked by $\ce{H}$). It is imaginable that all of these methyl groups are attached on one side of the chain only (the upper picture), or consistently "once on one, the next time on the opposite side" (the lower picture).  Not shown here, there may be a randomized distribution, too, where you can't predict on which side the next methyl group will be attached (not shown here) as an intermediate situation.

Such alterations in the molecular structure may alter how individual polymer molecules interact with each other.  It influences if the polymer becomes more crystalline, or more glass-like; and by the same token, it influences the melting point of the material.  Indeed, the reference states melting points of 130 degree Celsius (266 Fahrenheit), as well as 171 degree Celsius (340 Fahrenheit).

Now think about the production of the bottle: it is done by (blow) molding.  A manufacture aims to produce a high number of bottles per unit of time, and preferentially at a low cost to spend, for example to heat the polymer that it becomes flexible enough to form it.  So it is better to select a polymer with a lower melting point, and by admixing / blending with other polymers to polypropylene, the melting point probably may be lowered further.

On contrast, a manufacturer of a linings of dishwashers equally takes into account the processes while the dishwasher is used: chemicals (detergent), hot water during the washing, and heating to eventually dry the tableware.  So the selection prefers a polymer that tends to be more crystalline and hence exhibits a higher melting point.

You may encounter situations showing you that this comes with an disadvantage, too:  The more crystalline, high melting polypropylene liner often is more brisk than the more glassy, low melting polypropylene of your bottle.