I don't think it's gravity as that is well measured to the scale of nm/s^2 and any external localized field distortion would have been measured.

If I have to guess, it would be that the below Indonesia (where the bulge is according to the article) there is more radiative cooling happening either due to a thinner crust or higher concentration of more thermally conductive elements in the inner and outer core/mantle/crust. These effects can be localized because the soup from which the Earths' insides were made was "liquid" to a point and so an even mix of elements is highly unlikely even after accounting for billions of years of gravitational differentiation. For example, the outer core is the only technically liquid part and it still has an incredibly high viscosity (1000s to billions of times the viscosity of water [1]) and definitely you wouldn't call that a liquid in your day-to-day life.

[1] https://arxiv.org/abs/0709.3333

You don't necessarily need gravity to make something tilt, it can happen naturally during rotation. For instance, imagine spinning something that distorts asymmetrically, like a pencil with some silly putty attached to one side.

Alternatively, there's stuff like this: https://www.youtube.com/watch?v=1VPfZ_XzisU. Angular momentum's always conserved, but the axis of rotation might not be.

Tilt is with respect to our orbit around the sun and apparently mostly the moon is to blame for the tilt.

https://www.obliquity.com/info/meaning.html

I think it's a good question. What deformations would we expect if the Earth had captured a very tiny black hole?

The Sun and the Moon have a big enough impact to make the our planet spin, orbit the Sun, cause tides, etc.