The US President can receive clothing as a gift. They, along with the Vice President, are pretty much the only federal employees who can accept substantial gifts, even if they come from 'prohibited sources', as long as it comes from a member of the American public; receiving gifts from foreign governments/officials without the consent of Congress is explicitly forbidden by the Foreign Emoluments Clause of the Constitution.
Presidential Exemption
The Electronic Code of Federal Regulations §2635.204(j) states:
Gifts to the President or Vice President. Because of considerations relating to the conduct of their offices, including those of protocol
and etiquette, the President or the Vice President may accept any gift
on his or her own behalf or on behalf of any family member, provided
that such acceptance does not violate §2635.205(a) or (b), 18 U.S.C.
201(b) or 201(c)(3), or the Constitution of the United States.
The linked exceptions relate to the gifts being received in return for, for example, being "influenced in the performance of an official act", or to "do or omit to do any act in violation of the lawful duty" of their office.
General Rules
The rules on gifts from outside sources for federal employees that the above regulation exempts the President and his VP from are set out in the rest of Subpart B. Generally, a federal employee may not accept any gift if it may be seen by a "reasonable person" to question his or her integrity. In addition, gifts may not be accepted if they come from a 'prohibited source'.
A prohibited source is defined in §2635.203(d) as an individual or organization seeking official action by, doing business with, or conducting activities regulated by the employee's agency, or has interests that may be substantially affected by the performance or nonperformance of the employee's official duties.
So the rules exist to prohibit both actual corruption and actions that may be seen as corrupt. The President and Vice President are exempt from these regulations, according to the Congressional Research Service, because:
In promulgating its rules and exceptions, the Office of Government
Ethics has noted that: “The ceremonial and other public duties of the
President and Vice President make it impractical to subject them to
standards that require an analysis of every gift offered.”
Although the President and VP may accept gifts of this nature, they are required to report them if they are over a certain amount. The requirements for this are set out in 5 USC App 102: Contents of reports.
Possible Explanation
Let's now look at where this misconception may have come from. An AP article looked at who paid the bills for Michelle Obama's wardrobe while she was First Lady, and includes a quote from her press secretary:
Here's how Joanna Rosholm, press secretary to the first lady, explains
it:
"Mrs. Obama pays for her clothing. For official events of public or
historic significance, such as a state visit, the first lady's clothes
may be given as a gift by a designer and accepted on behalf of the
U.S. government. They are then stored by the National Archives."
So we see in this case that the clothes were, in fact, donated to the U.S. Government, not the First Lady herself (via her husband). This means that the gifts are handled differently; according again to the Congressional Research Service:
Since the President is not flatly prohibited from accepting gifts from
the general public, such a gift made to the President personally, and
accepted, may be retained by him when he leaves office. Gifts coming
to the White House that are not intended for the President or First
Lady personally, however, but rather are given with the intent to be
made for the “White House,” or otherwise made to the government of the
United States, and personal gifts not retained by the President or
First Lady, are catalogued, distributed, or disposed of by the United
States.
The article also states that the First Lady also has to be careful with accepting discounts:
Several designers who have provided clothes for the first lady
declined to discuss their arrangements. But given the prestige that
comes with dressing Mrs. Obama, it's widely thought that designers are
eager to cut the first lady a break. Former White House lawyers said
any discounts provided to the first lady would have to be in line with
what designers offer other top customers to avoid being considered
gifts.
This reinforces the point - the President and the Vice President may accept personal gifts or gifts on behalf of their family members, but the family members themselves receive no such exemption.
In addition, the Congressional Research Service does note that in practice, gifts to the President/VP are screened, categorized, and evaluated by the White House Gift Office, before being distributed.
If personal gifts to the President or First Lady are not to be
retained by them, they are generally recorded, tracked, and sent to
the National Archives and Records Administration [NARA] for courtesy
storage, and possible eventual use and display at a presidential
library.
Conclusion
Although the President/Vice President is subject to the general restriction enshrined in the Constitution which means they may not accept gifts from foreign governments without the consent of Congress, they and their family members, are not subject to the general restrictions set out by the Code of Federal Regulations on accepting gifts - unless they are given in return for preferential treatment, or solicited - even if they come from 'prohibited sources'.
The confusion displayed in the articles linked in the question may stem from the fact that in previous instances, clothes have been donated to the U.S. Government for use by the First Lady at public occasions, and have then subsequently been accepted by the National Archives. Finally, the Congressional Research Service includes an overview of the process in the Reagan White House:
While gifts from family and friends go directly from the White House
Gift Unit to the first family, it is simply impossible for the
President and First Lady to retain, or even view, most of the gifts
from the general public. The Gift Unit, therefore, sees to the
disposition of most of these items. Some are transferred to the
National Archives, and eventually join head of state gifts as part of
a presidential library museum collection.
