How is Lent observance reconciled with Matthew 6:16-18?

There are a few public acts that occur during Lent:

1. wearing blessed ashes
   According to Baltimore Catechism 3.1077, they

   are used to keep us in mind of our humble origin, and of how the body of Adam, our forefather, was formed out of the slime or clay of the earth; also to remind us of death, when our bodies will return to dust, and of the necessity of doing penance for our sins.

2. communal fasting and abstinence, which is a precept of the Church.

Both of these acts are a public profession of faith, and "We are obliged to make open profession of our faith as often as God's honor, our neighbor's spiritual good, or our own requires it." (Baltimore Catechism 1.326) "Every one therefore that shall confess me before men, I will also confess him before my Father who is in heaven." (Mt. 10:32).

^{Up until the mid-19th century in the U.S., even some public schools did not serve meat on Fridays because of sensibilities to Catholics.}

St. Thomas Aquinas, in his commentary on Matthew, says communal fasting and abstinence does not apply to Matthew 6:18:

That thou appear not to men to fast. Here is the reason [for the suitable manner of fasting]. [This reason] ought to be understood of individual fasting and not of common fasting.

There is nothing about Lent observance that requires it to be public.

My experience is very different from yours, and most people keep their Lent observances very private. I don't know what anyone is doing for Lent except for my immediate family (I have deduced one person's Lent observance, but even then it was something they kept as private as possible.)

I've been part of Lent in several churches, and in no case was there any encouragement to share your observances with others. I would go so far as to suggest that if the people you are with are making it public they may be doing it wrong.

(Note: None of the churches I've been involved with were Catholic, so it's possible there may be a difference in practice there. I will leave it to a Catholic here to write another answer if appropriate.)

How is Lent observance reconciled with Matthew 6:16-18?

Lent can be as private as on wishes. The public nature of Lent observance is generally limited the liturgical days of Ash Wednesday and other liturgical days in Lent. Our individual practices can and should remain between ourselves and God.

Apart from the only two days in which fasting and abstinence is required and the traditional days of observing abstinence on all Fridays of Lent, the vast majority of Catholics where I live, do not talk about what they actually give up for Lent!

Nothing is worse than vain glory!

Nothing irritates some persons more than hearing them state triumphantly that they gave such and such for Lent!

Do not be the show-off! Humility is the more conducive way to observe Lent.

I take St. Paul words to help me through Lent:

Each one should give as he has decided in his heart, not reluctantly or from compulsion, for God loves a cheerful giver. - 2 Corinthians 9:7

Even St. Benedict in in his Rule talks about the observance of Lent for his monks. He encouraged his monks to inform the abbot of what they were to observe for Lent. The others simply need not to know!

Chapter XLIX: Concerning the Observance of Lent

Although the life of a monk ought always to have a Lenten character, yet because few have the degree of strength requisite for that, we therefore exhort that at least during Lent he live his life with scrupulous care and that likewise during this holy season he do away with any departures from strictness that may have been permitted at other times: and this is then done worthily when we restrain us from all faults and give heed to prayer with tears, to reading and to heartfelt penitence and to abstinence. Therefore at this season let us betake to us, as some addition to the accustomed severity of our holy servitude, special prayers and abstinence from food and drink, so that each of his own free will, with joy of the Holy Spirit, may offer to God somewhat over and above the measure laid upon him; that is to say, let him deny himself in the matter of food, of sleep, of talking, of mirth; and let him look forward to holy Easter with the joy of spiritual longing. Let each one however confide to his abbot exactly what it is he is offering and let it be done with the help of his prayer and with his consent, because what is done without the consent of one’s spiritual father will not be accounted meritorious, but presumptuous and vain-glorious. Therefore it is with the abbot’s consent that all things are to be done.

In order to avoid the vain glory that St. Benedict speaks of, the monk should follow what St. Matthew says in his Gospel. Only his abbot and God knows what he is truly giving up for Lent.

Fasting to Be Seen Only by God

16 “Moreover, when you fast, do not be like the hypocrites, with a sad countenance. For they disfigure their faces that they may appear to men to be fasting. Assuredly, I say to you, they have their reward. 17 But you, when you fast, anoint your head and wash your face, 18 so that you do not appear to men to be fasting, but to your Father who is in the secret place; and your Father who sees in secret will reward you openly. - Matthew 6: 16-18