Probably not as a ready-made dual-purpose product, and it’s usually unnecessary to remove the handle. In photography, umbrella mounts normally clamp to the light stand and include a hole for the umbrella shaft, so the umbrella itself isn’t directly mounted by its handle.

A regular umbrella could be adapted, but there are tradeoffs. For shoot-through use, the material needs to be translucent white; most rain umbrellas are not designed for good light transmission or neutral color. Sheer waterproof material is uncommon, and ordinary fabric may affect color, though white balance can compensate somewhat.

If you want to experiment, a clamp-based solution is the most practical approach: use a grip clamp or super clamp to hold or position the umbrella shaft rather than modifying the handle. You could also DIY-modify an umbrella, but standard consumer umbrellas may not be sturdy enough for studio mounting compared with purpose-built photo umbrellas.

So: possible with clamps or DIY, but purpose-built photographic umbrellas plus a separate weather umbrella are usually the better solution.