Might not have been the same glue, but we had a similar tile under square linoleum type tiles. I used a green scrubby, clorox clean-up and very hot water with occasional abbrsive cleanser. We were told there is an adhesive remover at home depot but this worked fine with lots of scrubbing. Also, let the hot water puddle and soak for 10-15 min on an area first to soften the glue. After trying to repair the original floor, we ended up removing it and replacing with a similar new tile because of a big spread crack and slope at the toilet end of the room. Hopefully yours is a keeper! Here is a link that might be useful: album of bathroom renovation

Okay. I gave up, and called in the pros... They used a professional floor scraper (basically, a large razor blade on a long stick) to get the lino up, and then used a giant sander with 24 grit sandpaper to get the tar up. They also told me that it is NOT asbestos - just plain old tar. After 8 months of me and my neighbor and his friends scraping, scraping, scraping... it took the pros about two hours. It looks AMAZING. I am thrilled. Its only just been sanded - I'm waiting on the matching heart pine I ordered to patch it, and then they'll put the finish on it. It's costing me $1200 for scraping, sanding, and finishing, plus an extra $400 for the patch work (including ripping the 5" boards down to fit in my 3 1/4" floor). The heart pine boards to patch? $38 a box, times two boxes. And to tell you the truth? Totally worth it. And it should be done after three days of work. Three days! I have been up all night every night in the last week stressing about the darn floor - the idiots my neighbor got to work on it (supposedly pros) damaged it, and I was sure it was just ruined. Now I can finally get some sleep. That right there is worth the money.

I found this on https://www.gsa.gov/portal/content/113602 Removing Linoleum Adhesive From FloorsProcedure code:965001SSource:Ohj - Shirley Maxwell & James C. Massey, 9/30/92Division:FinishesSection:Resilient FlooringLast Modified:08/02/2016 REMOVING LINOLEUM ADHESIVE FROM FLOORS PART 1---GENERAL 1.01 SUMMARY This procedure includes guidance on removing the mastic adhesive left behind when linoleum flooring has been removed. Linoleum flooring was very popular from 1863 to 1974. It was used historically as an original floor treatment in some GSA buildings. Linoleum's chief ingredient was linseed oil, pressed from flax seed, which became a tough, elastic material when exposed to air. The process essentially consisted of mixing linseed oil and gum with ground cork or wood flour, and pressing it onto burlap or canvas. It was then glued down to concrete or wood floors using a mastic cement. See "General Project Guidelines" for general project guidelines to be used along with this procedureThese guidelines cover the following sections: Historic Structures Precautions Submittals Quality Assurance Delivery, Storage and Handling Project/Site Conditions Sequencing and Scheduling General Protection (Surface and Surrounding) These guidelines should be reviewed prior to performing this procedure and should be followed along with recommendations from the Regional Historic Preservation Officer (RHPO). PART 2---PRODUCTS 2.01 MATERIALS Clean, potable water Vinegar Citrus degreasing solvent such as "Limo Sol", or approved equal. Dry ice Chemical solvent such as isopropyl alcohol, paint thinner, xylol, lacquer thinner, or paint stripper. 2.02 EQUIPMENT Heat lamp or hot-air gun Putty knife or scraper PART 3---EXECUTION 3.01 ERECTION, INSTALLATION, APPLICATION NOTE: A waterproof glue is usually applied at the edges and seams of the linoleum; the center of the felt layers is usually attached with a water soluble paste. Soak the mastic area in hot water for 20-60 minutes or until soft.\ If this is not effective, try adding vinegar or high-strength citrus degreasing solvent to the water and then soak for 20-60 minutes until soft. If mastic still will not soften, try applying heat to the surface using a lamp or hot-air gun. -OR- Try freezing the mastic with dry ice to break the bond with the substrate. If no other methods are effective in softening the mastic, test a chemical solvent on the area. NOTE: Be sure to test in an inconspicuous area to determine appropriate chemicals and strengths before proceeding with cleaning. Scrape away the softened mastic using a putty knife or garden edger. END OF SECTION interesting................

Who knows why the staples are there but your guess may be correct. Getting them out will NOT be easy. Give your daughter a flat pry bar and a pair of pliers and let her have at it. ;-) Depending on the age of your house, that old flooring and/or it's adhesive may contain asbestos. How far out is your brother's tile work? You may want to have him take a look at things. He will have to do something with the old floor to lay his tile, so why do things twice?