If you say "winter 1998", is that too ambiguous?

There are people with the knowledge that "winter xxxx" is the winter from Dec. 21 XXXX to Mar. 20 XXX(X+1): https://miniwebtool.com/first-day-of-winter/?year=2020&location=1. In the words found on this site, "First Day of Winter 2020, Monday, December 21, 2020", the period of wintry season preceding December 21 does not include the first day of winter 2020 and that period must then be in the winter of 2019.
Here is a much more solid source confirming this logic: https://www.calendarpedia.com/when-is/winter.html.

According to that, the specific phrase "winter XXXX" or "winter of XXXX" should always be taken as referring to the later wintry part of the year XXXX enlarged by the early winter part of year XXX(X+1).

While winter can be defined in different ways, and with different degrees of precision, depending on the context in which the word is used and the purpose of the definition, it, on most definitions, denotes, in the Northern hemisphere, the season that straddles the end of one calendar year and the beginning of the next one. (This answer, like the OP's question is limited to the Northern hemisphere; an analogous question could be asked, and an analogous answer given, about the use of summer for a season in the Southern hemisphere.) Because of that, if one wishes to unambiguously refer to a specific winter, one has to say something like the winter of 1998-99. The phrases such as the winter of 1998, which mention only one of the calendar years, are bound to be ambiguous (when considered by themselves) between the winter that starts in that year, and the winter that ends in it.

Some people may be tempted to make the argument that the winter of 1998 should stand for the winter of 1998-99 on the ground that the season should be referred to by the year in which it begins. Others may be tempted to argue that it should stand for the winter of 1997-98, on the ground that the season should be referred to by the year in which the greater part of it is. Neither of these arguments is, however, universally accepted, so they are not of much help in determining what somebody actually meant by the winter of 1998.

Of course, as with most other ambiguities, the context can help one to discern which winter was meant by such a phrase. For example, if one is told that something took place in the autumn and winter of 1998, it is reasonable to assume that it is the winter of 1998-99 that was intended . On the other hand, in 'the winter and spring of 1998', it is likely that the winter of 1997-98 was intended. If one finds the winter of 1998 in a context that is mostly about Christmas, chances are that the phrase stands for the winter of 1998-99, but in a context that concerns the Valentine's Day, it is likely that the phrase stands for the winter of 1997-98.

In the absence of such disambiguating contexts, the phrases like the winter of 1998 are ambiguous, and wise, precise writers will avoid them, just like they avoid the formulations such as the midnight on the 10th.

Yes, it's ambiguous.

Aside from precision and which end of the northern winter might be intended, as the other answers note, there's also the question of whether the northern winter was intended at all. In the southern hemisphere, winter refers to the middle of the year.