Yalkut Yosef Hashkamat Haboker 2:1
Whenever someone conducts himself respectfully because he wishes to
show honor to God, and displays his awareness that God's Presence
fills the entire universe, he gains the title of "One Who Honors
God." The verse (Shemuel I 2:30), "I will give honor to those who
honor Me," applies to those people.
The way one dresses honours Hashem. It is associated with a sensitivity that Hashem fills the universe, a sensitivity that makes us desire to be respectable, because we are in the presence of the very great Master, Who desires us to be holy and modest.
Yevamot 78b:
David said: There are three distinguishing marks of this nation, the
Jewish people. They are merciful, they are bashful, and they
perform acts of kindness...They are bashful, as it is written: “And
that His fear shall be upon your faces” (Shemot 20:17)
Footnotes on Yalkut Yosef:
The responsa Igrot Moshe (Vol. III, Yoreh De'ah. #47:3: #68:4)
asserted that there is no halachic obligation for men to clothe their
entire bodies, but it is a highly recommended behaviour. Even when going to a restroom, one should not undress until one is about to sit
on the toilet...
In contrast to this, the responsa Mishne Halachot (Vol. VI, #2) ruled
that these instructions [to put on at least a robe before you even sit
up in bed] are an absolute requirement of the Talmud, Poskim, and
the Shulchan Aruch. The Ben Ish Chai (parashat Vayetze halacha
4) cited several Rishonim who listed modest dressing as one of the
Torah's commandments, whereas others listed it as one of the
outstanding traits of the nation of Yisrael, highly recommended by our
Sages. He concluded that we must take care to dress modestly since
there are those who ruled that it is a Torah obligation...
Suppose that it is extremely hot and a man who normally wears regular
shirts wishes to go about sleeveless. According to the Igrot Moshe,
this would be permissible. One is not expected to observe voluntary
halachot when it causes great discomfort. It may be, however, that
according to the Mishne Halachot, it would be forbidden even in
these circumstances...
The Ben Ish Chai (parashat Vayetze halacha 4) ruled that one must
make sure to lock the door when using the restroom, pointing out that
the Sefer Mitzvot Katan and other Rishonim counted modest
behaviour as one of the mitzvot of the Torah...
The Igrot Moshe stated that our Sages' definition of modest dress is
not that one must go about all day and night as one would be expected
to appear in the king's palace, or that one must dress in private as
one would dress in in the presence of ordinary strangers. Although we
must remain aware of God's Presence, the halacha demands that we dress
as if appearing in a king's palace only when praying the Amida, as the
Shulchan Aruch ruled (91:5). At other times, one is expected to dress
in a manner that is respectable, so that one would not be embarrassed
in the presence of ordinary people. He adds that when it is very
hot, there is no obligation to observe this standard either, since
everyone [including God] understands the situation. Regardless of the
circumstances, it is forbidden to expose one's genitals.
The Mishna Berura (2:1)... states that one must not expose any part
of the body that one normally keeps covered. One may reveal one's arm
up to the elbow and one's throat down to the top of the chest, he
specified. In countries where people always wear socks, one must put
them on while under the covers without exposing one's bare feet.
This gives an overview of the precise way halacha views the topic. It's a beautiful halacha to learn because it is about developing a sensitivity, rather than a just rigid set of laws (although of course there are rigid laws too). That is how the halacha is written and presented.
According to some opinions it is highly recommended to dress properly, with sleeves and trousers and socks, even when at home, and according to others, it is an outright obligation.
You asked specifically about a yeshivish family, and indeed a yeshivish family should be strict with this code of conduct (not taking on strong recommendations of Chazal would be like (l'havdil) a business owner not taking advice from his accountant, unimaginable if yeshiva taught them anything), although certainly not push in any way on people who do not wish to follow the opinion that it is halacha, or even upon the people who do not want to take the recommendation. Them doing so does not imply in any way that they are lesser in Hashem's eyes.
I'll finish with a story from yeshiva. We had someone who didn't dress very respectably, deliberately messy, untucked, and he was also generally quiet. He spoke to our Rosh Yeshiva once, who recommended he take it upon himself to start dressing smartly, and as a result he will feel better about himself, feel like a more respectable person, and indeed this is exactly what happened. He told me, and just about everyone how much of a real impact this had.
Even if one is not on the level to be bashful and fearful to undress in Hashem's presence, one will still move towards this by following the halachot, as when one feels more respectable, it opens gates and he is able to come closer to Hashem.
This effect also maintains respect in the home. Making it harder for others to find one respectable, which happens if someone lounges around in their underwear or topless, is putting up barriers to the close, holy intimate homes we are expected to produce. Respect between spouses is essential for shalom bayit. We can't deny the way Hashem made us, and He made us that we are much more able to be close with people we deeply respect. It goes without saying the importance of being respectable to one's children. It's not worth playing with that. Our home is our Mikdash Me'at.
If you want to get more information on this, I do recommend getting the Yalkut Yosef, the english Saka Edition is what I used for the above quotes if you prefer english. It is very good and the section is quite long, and really gives a good portrayal of how a Yirat Shamayim's outlook should be, as well as go into modern and local exceptions and the rules for that - some yeshivas may have different holy customs than the ones outlined in the above passage. I recommend learning the whole of the 1, all seifim, especially 1:8 on the halacha of being aware of Hashem's presence always, and all the footnotes.
