The term just means "son-in-law" and represents the Timurid/Mughal claim to legitimacy via marriage into the Golden Family (Chinggis Khan's descendants). Timur adopted the designation after marrying a Chinggisid princess to solve the problem of not descending from any particularly notable persons. He also claimed a spiritual descent from 'Ali; this effectively linked him to the two preeminent dispensations and sources of legitimate political authority in his era: the (apparently divinely sanctioned) conquests of Chinggis, and familial descent from the prophet through Fatima and 'Ali.

The reason the Timurids and the Mughals used the same term is that they were the same dynasty. There was no separation for the Mughals between the Timurids who had ruled in Central Asia and the Timurids who were currently ruling in India - Mughal is an outsider's term and draws a distinction that was not made at the time. The Timurids in Delhi continued to talk about/dream of the reconquest of their ancestral lands in modern Afghanistan for a surprisingly long time, considering how successfully things were going for them in India. They continued to perceive/describe themselves as Central Asian foreigners in a strange land for hundreds of years after their arrival, long past when you might expect them to have considered themselves natively Indian.

Here is one of many examples of how the Mughals continued to present their descent from Timur as the source of their legitimacy.