Well the short answer is yes but here is the kicker... so does all site-built homes, mobile homes/manufactured homes/modular homes, RVs/campers, and Tiny Homes. Formaldehyde is used in a ton of day to day products such as furniture, carpeting, plywood, particle board, OSB, paneling, household cleaners, wall board, shelves, cabinets, countertops, and so on. By looking at the list of things that I just wrote above, it is obvious that every kind of house out there will have formaldehyde in them to some degree. The more of these products that a home has in them, the more formaldehyde exposure there will be naturally.
Mobile Homes or more commonly called Manufactured Homes were given a bad rap right after FEMA purchased thousands of mobile homes and RVs for people to live in after the Hurricane Katrina flood. Most of those homes were built and shipped straight to their locations, setup and were immediately moved in to. Generally, mobile homes sit on dealer lots for a while before they are sold and moved in to and therefore have a chance to air out before they are lived in. Formaldehyde didn't cause anyone to die or anything back during the whole FEMA house dilemma but it did cause minor irritation to the eyes, throat, and sometime lungs. Those particular homes were also made very cheaply and used a lot of paneling for the walls on top of everything else in the homes that were already made with formaldehyde. The combination of all of those products and no time to air out ended up being a bad combination. Homes these day do still use a lot of those same products, but they have started using less paneling and have shifted more to sheetrock walls which helps out on the formaldehyde amounts. Also, some manufacturers have carpeting now that is formaldehyde free as well. These positive changes will only help to improve these homes in the future and lower their formaldehyde exposures.
When compared to site-built houses the main real difference is the amount of time that it takes to build them and the amount of time they have to air out. Site-built homes take 4 months to a year to build and are constantly airing out the entire time, making their formaldehyde exposure lower once the homeowner is able to move in. Manufactured Homes are built in less than 2 weeks so all of those products are releasing formaldehyde at the same time, making them have a stronger formaldehyde exposure if a homeowner were to move in immediately after being built. The best thing to do if ordering a new manufactured home or even buying one off of a lot is to open the windows and just let the air out as much as possible. The longer that this is done the more the formaldehyde can escape. Also, getting the A/C turned on as quick as possible will help as well. Homes without A/C that sit during the summertime (on a dealer's lot) tend to have a strong formaldehyde appearance as well since they get super hot and have no air flow through them.
Basically, don't worry about the formaldehyde exposure in a mobile home. You will get it no matter what you are living in. Mobile homes just ended up getting a bad rap a long time ago and that stigma is still sticking with them today. If you do enough research, you will find that living a formaldehyde free life is pretty much impossible!