Yes. The ACLU actually is larger and protects more than one specific amendment. Additionally, the portion of the constitution that applies to the rights of the people is relatively small compared to the the full body. Generally, the first 10 amendments are written as protections of individual rights (known as the Bill of Rights), and all other amendments are definitions of who those rights apply to, or changes to government's structure or authority.
In the Bill of Rights, three are safeguards of liberty (Amendments 1-3), four are safeguards of justice (4-8), and two are are protections of unenumerated rights (fancy language for "things we were too lazy to bother to write") and reserved powers (9-10).
All other amendments to the constitution are further restricting the government in laws with respect to enfranchisement (who can vote) and protection of rights, with the former being the bulk of the Civil Liberties Amendments. Of the enfranchisement rights, you have 15th, 19th, 23rd, 24th, and 26th. The protection of rights is only the 13th and 14th amendment. Enfranchisement offers no further liberties but bars the government from making laws that would prevent people from voting based on: Race, color, and former slave or servant class status (15), sex (19), D.C. residency (for the President of the United States only) (23), Economic Status (24), and sets Age of Enfranchisement as 18 (26). (This was previously set by the states. The reason had nothing to do with setting a universal age which was 21, but to make enfranchisement align with draft age... prior to this, one could be required to fight in a war without even having a chance to vote on the people who can call for a draft.)
The 13th Amendment prevents the government from allowing slavery and empowers Congress to penalize slavery in the United States. The 14th amendment patches up the legal fuzziness that was largely responsible for the Civil War.
In U.S. constitutional law, the U.S. Constitution is defined as rules for the Federal Government, not the state governments (the powers of the state in Article IV are things a state will give up to be part of the federal government and benefits), which each have their own unique constitution with their own rules of government. To enforce this, the 9th and 10th amendment, when combined, created the following situation:
- If the Constitution says the Federal Government can do something, then the States or Individuals may not do it.
- If the Constitution says the Federal Government cannot do something, then the Federal Government may not do it.
- If the Constitution doesn't say the Federal government can or cannot do something either way, then the Federal Government cannot make rules that stop the states or individuals that want to do it.
The problem with this is that the Constitution binds the Federal Government, but not the states. Thus under this rule, the First Amendment blocks laws restricting political speech, but states may restrict political speech. In the case of slavery, the Constitution had several lines that discussed it (in Article IV when discussing representation with respect to censuses), but didn't ban states from allowing it, and didn't empower the Feds to regulate it. Thus, states could allow or abolish it to any degree they wanted, and there was nothing the Feds could do about it. (They could ban importation of slaves, because international trade was a Federal power, but they couldn't restrict the sale of slaves or their domestically born children already in the United States.)
The 14th Amendment patched this by changing the rules of the Constitution so that if the Constitution explicitly says in writing that Congress cannot do something, then the states definitely cannot do it either. This is important as the 13th Amendment bans Congress from enacting laws that allow slavery to exist and must enforce no slavery. Because of the nature of the Bill of Rights being written on limits of the Federal Government's authority (Congress may Pass no Laws, The rights of the people shall not be infringed, the executive cannot violate the rights without following these due process, the judiciary must have just cause), the 14th Amendment applies Amendments 1-9 to the states, by amending Amendment 10.
In theory, the ACLU does protect Second Amendment rights, but they also protect everything else as well. The First Amendment is very broadly interpreted and generally covers most of every liberty. (It has five specific limits to the government. The Second Amendment has just one.) The Third Amendment is rarely violated, so there's little case law. The Second Amendment is more controversial than the Third, but is more narrow than the First. The controversy is that it does allow for regulation, so long as the regulation does not ban gun ownership outright (under case law read).
What the Feds can do with that and what they can't and what the states can and can't is a huge spectrum if "Ban 'em all" is off the table.
There are also specialized lobbies in the United States. For example, there are a few lobbies that protect just the religious freedom aspect of the First Amendment (even from the point of view of a specific religion, such as CAIR, which is Islam-specific advocacy group, or the Anti-Defamation League, which was formerly Judaism-specific). Other groups may only support the spectrum of journalism rights, and not care about rights of protest.
The NRA specializing in the Second Amendment is narrow because there's one specific limit in it. A First Amendment advocacy group can be general or narrow into one of the five limits, each of which has specifically defined case law. And no one really advocates for the 25th Amendment rights of the people, because the person it affects is the President, and it's really about who gets his job if he dies, so hardly a legal case, but a lot of fun for fiction writers to play with the what ifs.