Nothing prevents people from naming Mosques after certain people, and the name could be different from the person who built the mosque, for instance, to commemorate the name of a scholar, ruler or reformer, and he could be the one who built the mosque and named it after himself.
As for immortalizing the names of people with regards to mosques, there is nothing preventing that so long as the person deserves that, and the intention of the person naming the mosque is good.
As for writing the name of the person who built the mosque on the mosque, then it also goes back to what his intention was. If it was simply to distinguish it from others and it is easier to find, or do necessary procedures then there is no objection to that, however if it is done with pride, and to show off then it is not allowed. There are many texts regarding our obligation being solely for Allah SWA and the prohibition of showing off. Similarly, the well known hadith (Verily actions are judged with there intentions).
If the intention behind naming the mosque after the person who built the mosque is to invoke interest in rich people to get them to participate in religious projects such as building mosques, then perhaps he is worthy of appreciation and respect, and Allah SWA is the one who judges the intention. This is similar to the competition between the companions on financing the Expedition of Usra (Tabuk), and the efforts of some to ensure that their contribution was more than the others. The Prophet ﷺ complimented Uthman who contributed a lot, and Allah SWA says in these circumstances (If you give (your) alms openly it is well and good (in itself). But if you keep them secret and give them to the needy it is better for your own selves and He will (thereby) acquit you of some of your evil deeds. Allah is fully aware of what you do.) Al Baqarah: 271
And there is no connection between naming the mosque after certain people and the verse (And the mosques are for Allah). The meaning of this verse is that mosques are the houses of Allah SWA, no one else is worshiped in them apart from him, like the idolaters used to worship the idols around the mosque in Makkah, and it is the house of Allah SWA.