What al-islam.org writes from the shi'a perspective seems to be broadly the same from a sunni perspective.
islamweb.net (which I've read is an affiliate of islamQA, but I couldn't find any evidence of that) says
Irrespective how rich the wife is, the duty to give lodging, boarding,
clothing and look after the financial aspects of the wife remains that
of the husband.
islamQA just lists the right to lodging as one of the financial rights of the woman and offers no caveats with regard to when those rights might not exist.
"Reliance of the traveler", the shafi'i legal manual of fiqh, talks about the wife's financial support in section m11. Specifically, section m11.8 says
The wife is entitled to housing of the same quality as that of similar women. (O: The standard of housing depends on the wife herself, while the standard for her clothing and support takes the state of the husband into consideration. The difference is because the expenditures for her support and clothing become her own property and are not merely for her use, while housing is solely for use (N: meaning that while she can take compensation in place of food or clothing and buy some other kind, she cannot rent a different house). In any case, she is obliged to stay in the lodgings her husband arranges for her.)
("O" is a commentator on the Arabic original, "N" is the translator into English, Shaykh Nuh Ha Mim Keller.)
Note that this answers this related question : what if the husband insists that they live in a house that he rents/owns instead of her house that she offers? This seems to be clear-cut: the husband has a legal right to decide where his wife lives.
Section m11.9 talks about the conditions for support to be obligatory on the husband:
The husband is only obliged to support his wife when she gives herself to him or offers to, meaning she allows him full enjoyment of her person and does not refuse him sex at any time of the night or day.
She is not entitled to support from her husband when:
(1) she is rebellious (nashiz, def: m10.12(N:)) (0: meaning when she does not obey him) even if for a moment;
(2) she travels without his permission, or with his permission but for one of her own needs;
(3) she assumes ihram for hajj or 'umra (def: j3);
(4) or when she performs a voluntary fast without her husband's permission (0: though if he allows her to fast and does not ask her to break it, he must provide her support).
I omitted the referenced definitions because the gist of the conditions is clear in as much as they pertain to the question.
So far, this establishes that having a house of her own or being too wealthy is not one of the conditions under which the husband is not obligated to provide lodgings for her. About the matter of voluntarily providing lodgings herself, there is only this related consideration to be found in section m11.13:
The wife is entitled to annul their marriage whenever the husband is unable to provide her with the support obligatory for a nonaffluent person to pay (def: m11.2) and provide clothing or housing for her.
If she wishes, she may choose to bear with him (0: supporting herself with her own money), and it (0: the amount the husband is unable to pay) remains a financial obligation that he owes her. (0: If she does not wish to tolerate his financial incapacity, she cannot annul the marriage by herself, but must establish her husband's inability to support her before the Islamic judge, who annuls the marriage or allows her to do so, since he is the one who judges the matter (A: and if there is no judge, she has two persons (def: 021.4) decide)).)
Since the wife is allowed to provide for lodgings herself if the husband neglects or is unable to provide for her lodgings, it should be allowed if they agree on it as well. Her agreeing to live together in her house after he offered to provide lodgings for her may be considered waiving her right to lodgings, and - in the context of the last blockquote - waiving her right to what he would owe her as compensation for not providing for her.
The chapter on maintenance of the wife (40: nafaqat) in the mukhtasar of al-Quduri, one of the foremost legal manuals in hanafi fiqh, also makes no mention of the wife's affluence somehow removing the husband's obligation to provide for her, lodgings or otherwise. It does mention other reasons for that obligation becoming defunct, such as the wife deserting the husband, so - as with reliance of the traveler - it seems reasonable to assume that if her affluence was a condition under which that obligation becomes defunct, it would have been mentioned.
Note that all sources talk about rights, and this is meant as rights in the legal sense that can be sued for; marriage in Islam is a contract that gives the spouses enforceable rights over each other, not a sacred covenant as for example in Christianity.
That much about fiqh, it seems perfectly legal to move into her house together if they agree on it, and she might be waiving her right to lodgings for as long as they live there together; the same conclusion can probably be drawn for her other financial rights. In as far as cultural practices among Muslims were the intended topic of the question, that would be off-topic here.