Since the similarity between them is, as you point out, more thematic than linguistic, it may be useful to examine if this theme exists before and outside of Amos and Isaiah.
Earlier Examples of this Theme
The Psalms and Proverbs offer several instances that seem to be the beginnings of these sentiments:
The one who offers thanksgiving as his sacrifice glorifies me;
to one who orders his way rightly
I will show the salvation of God!”
16 For you will not delight in sacrifice, or I would give it;
you will not be pleased with a burnt offering.
17 The sacrifices of God are a broken spirit;
a broken and contrite heart, O God, you will not despise.
18 Do good to Zion in your good pleasure;
build up the walls of Jerusalem;
19 then will you delight in right sacrifices,
in burnt offerings and whole burnt offerings;
then bulls will be offered on your altar.
To do righteousness and justice
is more acceptable to the LORD than sacrifice.
Taking the position as stated that Isaiah 1 was written by Isaiah ben Amoz, he would have been contemporary with both Hosea and Amos. Hosea also seems to echo the Psalms and Proverbs:
For I desire steadfast love and not sacrifice,
the knowledge of God rather than burnt offerings.
But when prophesying against Israel he expands this description to the point where it matches Isaiah 1 quite closely, even more-so than Amos 5.
And I will put an end to all her mirth,
her feasts, her new moons, her Sabbaths,
and all her appointed feasts.
This does not quite include the explicit sentiment of weariness and disgust we see in Isaiah 1 and Amos 5, but in the larger context of Hosea we see that the punishment is not removing them as one corrects a child by taking something they love, but rather God is removing his blessings from them. The Sabbath and feasts and sacrifices are for Israel's benefit, not God's, and He is declaring them no longer worthy of such blessing: Hosea 2:9
Therefore I will take back
my grain in its time,
and my wine in its season,
Compare this to Amos 5:
21 “I hate, I despise your feasts,
and I take no delight in your solemn assemblies.
22 Even though you offer me your burnt offerings and grain offerings,
I will not accept them;
and the peace offerings of your fattened animals,
I will not look upon them.
Amos and Isaiah
It is quite possible that Isaiah did have knowledge of both Amos and Hosea. Internally, all three indicating their preaching in the reign of Uzziah.
Isaiah would likely be aware of Amos and what he preached even if he was not of Hosea, because Amos was actually from Judah himself (Amos 1:1), though he was sent to the Northern Kingdom of Israel.
However, as pointed out by Tau, if we are to believe that Isaiah ben Amoz wrote Isaiah 1 then we can be sure that he would have had access to any extent writings, possibly more than we have today:
2 Chronicles 26:22
Now the rest of the acts of Uzziah, from first to last, Isaiah the prophet the son of Amoz wrote.
The vision of Isaiah the son of Amoz, which he saw concerning Judah and Jerusalem in the days of Uzziah,...
Informed by this evidence, when we compare Isaiah 1 with both Hosea 2 and Amos 5, we see a clear synthesis of the two along with earlier Psalms and Proverbs.
Isaiah 1:13-14, 17
13 Bring no more vain offerings;[Psalm 50 & Amos 5]
incense is an abomination to me.
New moon and Sabbath and the calling of convocations— [Amos 5 and Hosea 2]
I cannot endure iniquity and solemn assembly. [Amos 5]
14 Your new moons and your appointed feasts [Hosea 2 & Amos 5]
my soul hates; [Amos 5]
they have become a burden to me; [Amos 5]
I am weary of bearing them. [Amos 5]
17 learn to do good; [Proverbs 21]
seek justice, [Proverbs 21]
bring justice to the fatherless,
plead the widow’s cause.
I admit a lack of knowledge of the Hebrew itself, but the similarities are evident in the English translation. And an untrained examination of the Hebrew reveals that the same root Hebrew words are being used where the English translates them the same:
Phrases such as "solemn assemblies" (עֲצָ֫רֶת) and "appointed feasts" in conjunction with "new moons" (חָדְשָׁ֣הּ) and the call to "do good/righteousness" (מִשְׁפָּ֖ט) in Isaiah are nearly exact matches to the other passages.
Isaiah most likely had access to all the writings available in his time, as we see in 2 Chronicles 26. In addition, Isaiah 1 shows similarities in key words and phrases, and theme contained in Psalm 50, 51, Proverbs 21 and, in particular, Amos 5 and Hosea 2.
The largest issue left to be solved is whether Amos and Hosea would even have been written down for Isaiah to read. But whether it was written, or whether it was simply hearing each other's preaching, it is difficult to believe that three major prophets operating within 30 years would not be familiar with the details and themes in each other's messages.
Some may want to claim that they wrote independently in order to bolster their belief in the common inspiration of scripture by the Holy Spirit. However, there are plenty of reasons and examples to point to the consistent messages found in scripture without forcing us to ignore the evidence in this case.