Any marriage that took place in Ireland after 1864 would have been civily registered. The General Registry Officewww.groireland.ie
look for criteria.
A marriage certificate will probably give you both his and her father's name, place of marriage, date.
Leitrim belongs to a not for profit History Foundation and has both birth and marriage records for a fee. His name I think is fairly unusual. Before 1864 birth records were in the Parish and this is practically the only way to access them. Rootsweb/Ancestry do have some birth records.
A birth record would give you both mother and father's names and usually mother's maiden name, place/parish/date.
A Thomas Morahan appears in Griffith Valuation 1856
tenant of Wm. S. Guiness property 6a Townland Keenaghan
house, outbuilding and land 445 acres 3 roods 31 perches
The other names are Patrick, John, Michael & Bernardwww.askaboutireland.ie
In the Directories is just 1 Morahan:
Bernard Morahan, Carrick on Shannon - coach painter, Bridge St.
The only Tosney in Griffith Valuation for all of Ireland
are 2 in Sligo 1858 (bordering county)Daniel and Phillip in Parish Kilmacallan
There are 5 Morahan in the Leitrim phone book but no Tosney's.
Both Sligo and Leitrim belong to a History Foundation that has birth and marriage records. Manorhamilton records start in 1840. After 1864 you should be able to find records
I would say it is the correct marriage:
Thomas Morahan married Oct-Dec, 1889 Manorhamilton
LDS Vol.2 Pg. 215 #4179388
Annie Tosney married Oct-Dec, 1889 Manorhamilton
LDS Vol. 2 Pg. 215 #4179388
Usually a match in the Later Day Saints microfilm would confirm your findings. These films can be seen at most LDS Heritage Centers near you.
Most men got married age 20-25 and there are a few births
on the above website.