What is the comparison between themes for Lady Windermere's Fan and An Ideal Husband?

This is a great question because there are, indeed, plenty of similarities between the two plays.
Both plays deal with the typical themes of the traditional comedy of manners:

a preoccupation for looking "proper" and showing off excessively good manners
keeping up appearances
superficiality vs. reality
hypocrisy
class differences
climbing the social ladder
duplicity
blackmail

The situations are similar as well.
In Lady Windermere's Fan the key problem is keeping the self-righteous Lady Windermere from learning about her true origin. Unbeknownst to Lady Windermere, her real mother is Mrs. Erlynne. The latter is not welcome in society because she abandoned her husband and baby daughter (Lady Windermere) to elope with another man.
As such, Mrs. Elynne attempts to get back in society by planning to disclose the truth to her daughter and by using her daughter's husband's good influence to ingratiate herself with the right "set" again. It is then that she finds out that her daughter is about to commit an act similar to what she herself did. Mrs. Erlynne decides to make a sacrifice to change Lady Windermere's mind, thus saving her daughter's reputation. As such, she cannot disclose who she really is. In the end, Erlynne's emotions took the best of her, and she ends up leaving altogether.
In An Ideal Husband, politician Sir Robert Chiltern attempts to prevent the shady Mrs. Cheveley from disclosing publicly how he made his fortune; Rather than making the money legitimately and incorruptibly, the way that the public who trusts him perceives, we learn that the "righteous" and "ideal husband" Sir Robert was into selling state secrets and doing insider trading, which is a white collar crime by our modern standards. If this type of crime is frowned upon by our society, imagine how big a scandal this would be from the perspective of the self-righteous Victorian society.
In return for her silence, Mrs. Cheveley simply wants Sir Robert to support a scheme project to build an Argentine Canal. He has publicly rejected the project, but now he is faced with the dilemma of having to backpedal his position in order to avoid public shame. In the end, love also wins, Mrs. Cheveley is exposed for what she is, and everyone decides to erase the past and move forward.
The love letters:
In both plays, the mounting pressure makes the wives seek succor in the dandies. Lady Windermere writes a letter to Lord Darlington proposing to elope with him. Similarly, Lady Chiltern writes a pink note to Lord Goring telling him to wait for her at his home, a very daring social faux pas. None of the notes end up making a big difference in either play.
Characters:
In An Ideal Husband we find the same archetypes as those in Lady Windermere's Fan.

Lord Windermere, the anguished husband in Lady Windermere's Fan, is Sir Robert Chiltern, the other anguished husband in An Ideal Husband.

The semi-ornamental wife archetype of Lady Windermere is Lady Chiltern in An Ideal Husband.

The villain, in this case, the blackmailer that is Mrs. Erlynne in Lady Windermere's Fan is Mrs. Cheveley in An Ideal Husband.

Mrs. Erlynne blackmails for money and position as she threatens to reveal that she is the real mother of Lady Windermere.
Mrs. Cheveley blackmails Sir Robert by demanding that he reverses a political decision he made regarding a project to build an Argentine Canal.
Both the husbands and wives in both plays face potential social shunning and public embarrassment.
Also, both plays feature the one archetypal character that appears in every one of Wilde's works: the wayward, amoral, bon vivant dandy. In Lady Windermere's Fan this character would be Lord Darlington. In An Ideal Husband the dandy is Lord Goring.

Finally, the ending of both plays is similar: they all overcome the obstacles and move on happily into the future.
For more information on Wildean plays, check out the Cambridge Companion to Oscar Wilde and you will learn a lot more about differences and similarities among Wilde's plays.
https://www.cambridge.org/core/books/cambridge-companion-to-oscar-wilde/7A1EC1EC30E5EF6785639CAD1B6A8A9D